+      * 


-i 


K         4- 


X 


THE-  ELEMENTS  =0^=  ■Toe- 
differential  AND  INTEGRAL 

' CALCULUS ^ 


BASED  ON  KURZGEFASSTES  LEHRBUCH  DER 
DIFFERENTIAL-  UND  INTEGRALRECHNUNG,  Von 
W.  N ERNST,  o.  o.  Professor  der  physikal.  Chemie 
A.   D.    Univ.    Gottingen,  und  A.   SCHONFLIES,  A.  o. 

Professor    der    Mai-hematik    a.    d.    Univ.    Gottingen 


BY 


J.  W.   A.  YpUNG 


assistant  professor  of  mathematical  pedagogy  in 

the    university    of   CHICAGO 

AND 

C.   e.   LINEBARGER 

instructor  in  chemistry  and  physics  in  the 

lake   view    high    school,    CHICAGO 


NEW    YORK 
D.    APPLETON    AND    COMPANY 

1900 


^'- 


Qft303 
V7 


Copyright,  1900 
By   D.   APPLETON   AND   COMPANY 


\03^o  S 


GtHtRl^^ 


PREFACE 

The  Differential  and  Integral  Calculus  is  of  the  lirst 
importance,  whether  regarded  from  the  standpoint  of  pure 
mathematics  or  from  that  of  the  natural  sciences. 

Without  an  elementary  knowledge  of  this  subject,  even 
the  names  of  the  modern  developments  of  pure  mathematics 
are  inexplicable  jargon,  while  an  introduction  to  the  Calculus 
suffices  to  open  to  one  the  possibility  of  a  vague  apprehen- 
sion of  the  problems  of  advanced  research  in  mathematics, 
and  of  the  results  which  the  industry  and  acumen  of  the 
mathematicians  of  the  day  are  continually  producing,  some- 
what similar  to  that  which  he  may  have  of  the  problems 
and  results  of  chemistry,  of  Latin,  of  history,  of  biology. 
To  the  cultured  man  elliptic  integrals,  gamma  functions, 
differential  equations  should  be  terms  equall}^  intelligible 
with  the  carbon  compounds,  the  subjunctive  mood  in  indi- 
rect discourse,  feudalism,  protoplasm.  An  elementary  ac- 
quaintance with  the  Calculus  makes  this  possible  and  brings 
mathematics  within  the  range  of  the  interest  which  one  who 
leads  an  intellectual  life  feels  in  the  varying  phases  and 
achievements  of  the  intellectual  activity  of  the  day.  A 
branch  of  study  to  which  so  large  a  share  of  the  time  of 
education  is  given  up  as  is  devoted  to  mathematics  should 
surely  not  be  dropped  just  when  that  subject  is  reached 
which  in  a  few  months  would,  in  a  sense,  round  out  the 
attainments  of  all  the  previous  years  of  work,  and  which 
offers  a  point  of  vantage  for  a  general  survey  of  the  field 
of  mathematics  such  as  can  nowhere  earlier  be  found. 

iii 


IV  PREFACE 

From  the  point  of  view  of  the  natural  sciences  the  subject 
is  equally  important.  From  its  very  discovery,  the  Calculus 
has  been  the  indispensable  handmaid  of  the  physicist ;  for 
him  its  most  complicated  machinery  has  been  put  into 
motion  ;  its  most  formidable  engines  have  even  been  devised 
especially  for  him.  But  the  chemist  is  now  also  calling 
upon  the  Calculus  for  aid.  The  day  has  come,  "when  a 
sign  of  differentiation  or  integration  must  cease  to  be  an 
unintelligible  hieroglyphic  for  the  chemist,  if  he  does  not 
wish  to  run  the  risk  of  losing  all  comprehension  of  the 
development  of  theoretic  chemistry.  For  it  is  a  fruitless 
labor  to  squander  page  after  page  in  a  vain  attempt  to  ex- 
plain that  which  one  equation  makes  perfectly  clear  to  him 
who  is  initiated  into  the  mysteries  of  the  Calculus."*  A 
similar  day  is  approaching  for  all  the  natural  sciences.  In 
fact,  the  Calculus  and  the  mathematical  formulation  of  the 
phenomena  of  nature  are  inseparable. 

With  these  ideas  of  the  importance  of  the  subject,  one  of 
the  authors  of  the  following  work  has  for  several  years  been 
giving  courses  of  instruction  in  the  elements  of  the  Calculus, 
including  a  broad  survey  of  its  principles  and  methods,  and 
a  brief  sketch  of  its  ramifications  throughout  modern  mathe- 
matics, but  excluding  the  more  complicated  problems  and 
the  more  difficult  computations  which  should  be  taken  up 
in  the  detailed  treatment  that  is  necessary  as  a  basis  for 
further  mathematical  study. 

In  consonance  further  with  the  view  of  Herbart,  as  quoted 
by  Klein,f  that  mathematics  is  uninteresting  to  five-sixths  of 
the  students  unless  it  is  brought  into  direct  connection  witli 
the  applications  ;  and  of  Clifford, J  that  "  every  connection 

*  Jahn,'  Grundriss  der  Elektrochemie,  quoted  in  the  preface  to  Nernst- 
Schonflies,  Differential-  und  Integralrechnung. 

t  Klein,  Nichteuklidische  Geometrie  (Lithographed  lectures),  I.,  p.  362. 
X  Clifford,  Common  Sense  of  the  Exact  Sciences,  p.  257. 


PREFACE  V 

between  two  sciences  is  a  help  to  both  of  them,"  applications 
of  the  Calculus  were  made,  as  occasion  offered,  to  problems 
of  the  natural  sciences  as  well  as  of  pure  mathematics;  but 
the  lack  of  a  text-book  written  in  the  same  spirit  was  felt 
to  be  a  hindrance  to  the  attainment  of  the  best  results. 

Early  in  1896,  the  w^ork: 

Kurzgefasstes  Lehrbuch  der  Differential-  und  Integralrechnung  mit 
besonderer  Beriicksichtigung  der  Chemie,  von  W.  Nernst,  o.  b.  Professor 
der  Physikal.  Chemie  a.  d.  Universitat  Gbttingen,  und  A.  Schbnflies, 
a.  0.  Professor  der  Mathematik  a.  d.  Universitat  Gbttingen, 

which  appeared  in  the  latter  part  of  1895,  was  brought  to 
his  attention  by  his  colleague  in  the  present  work.  The 
latter,  a  pupil  of  Nernst,  had  been  duly  authorized  to  trans- 
late the  German  book  into  English,  and  had  made  some 
progress  in  doing  so.  While  the  German  work  was  intended 
primarily  for  chemists,  it  appeared,  even  in  this  form,  better 
suited  for  use  as  text  in  the  courses  in  the  Calculus  just 
mentioned,  than  any  available  work  in  English  ;  and  it 
seemed  possible  to  add  to  its  efficiency  for  this  purpose  by 
alterations  having  as  aim  to  enlarge  the  mathematical  con- 
tents, to  increase  mathematical  rigor,  and  to  adapt  the  style 
of  presentation  to  American  methods  of  instruction.  Ac- 
cordingly the  work  of  preparing  in  collaboration  a  transla- 
tion, revised  and  adapted  for  use  in  American  Colleges  and 
Technical  Schools,  was  undertaken  by  the  present  writers. 
It  was  thought  that  this  could  be  accomplished  by  additions 
which  could  readily  be  indicated  by  some  distinctive  mark, 
leaving  the  original  in  the  main  intact.  The  actual  work, 
however,  gradually  made  it  apparent  that  the  alterations 
which  seemed  desirable  were  so  serious  that  the  German 
authors  could  no  longer  be  held  responsible  for  the  matter 
in  its  new  presentation.  The  alterations  have  been  so 
numerous  and  so  interwoven  with  the  whole  fabric,  ranging 
from    changes   in   phraseology   to   the   adoption   of    a   new 


VI  PREFACE 

method,  from  the  most  trifling  omissions  and  additions  to 
the  omission  and  addition  of  whole  chapters,  that  though 
the  present  work  is  most  closely  based  upon  the  valuable 
work  of  Professors  Nernst  and  Schonflies,  it  is  but  just  to 
the  latter  that  the  present  writers  bear  the  entire  responsi- 
bility for  the  work  as  here  presented.  The  last  chapter, 
"The  Differentiation  and  Integration  of  Functions  found 
Empirically,"  is  simply  a  translation  of  the  corresponding 
chapter  of  the  German  text ;  otherwise  only  the  present 
writers  are  to  be  held  responsible  for  whatever  defects  may 
be  found  in  the  following  work,  while  its  merits  are  to  be 
ascribed  in  a  very  large  measure  to  the  German  work  upon 
which  it  is  based. 

In  the  topics  taken  up,  and  in  the  extent  of  their  treat- 
ment, the  German  work  has  served  as  model ;  a  very  great 
part  of  what  appears  (especially  the  presentation  of  those 
topics  which  have  long  since  become  the  common  basis  of  all 
elementary  works  on  this  subject)  is  a  translation,  more  or 
less  close,  from  the  German  Avork ;  the  distinctive  feature  of 
the  latter,  viz.  the  continual  use  of  illustrative  examples 
from  the  natural  sciences,  is  likewise  a  characteristic  feature 
of  the  present  work.  With  a  few  exceptions,  the  illustra- 
tions of  this  sort  used  in  the  German  text  have  been  retained, 
and  a  number  of  additional  ones  introduced.  But  here,  and 
wherever  necessary,  the  mode  of  presentation  has  been  radi- 
cally changed.  The  German  authors,  after  using  the  method 
of  limits  to  establish  the  fundamental  rules,  soon  introduce 
the  method  of  differentials  and  make  very  considerable  use 
of  it  thereafter.  Though  this  method  may  be  satisfactory 
from  the  physico-chemical  standpoint,  it  seemed  that  in  a 
mathematical  text-book  the  method  of  limits  should  be  used 
exclusively. 

The  writers  believe  that  the  work  as  herewith  presented 
is  the  first  elementary  American  presentation  of  the  Calculus 


PREFACE  Vll 

in  which  this  is  done.  Even  those  writers  who  introduce 
the  subject  by  the  method  of  limits,  usually  take  up  the 
method  of  differentials  sooner  or  later,  and  to  a  greater  or 
less  extent.  We  regard  this  as  decidedly  inadvisable.  We 
believe  that  even  with  methods  of  equal  rigor,  the  beginner 
is  on  the  whole  more  confused  than  helped,  if  a  subject  is 
presented  to  him  for  the  first  time  according  to  two  or  more 
different  methods.  In  our  subject,  the  methods  do  not  stand 
on  the  same  plane  as  to  accuracy.  If  a  logically  sound 
"  method  of  differentials  "  is  set  up,  it  is  no  longer  a  distinct 
method,  but  only  a  different  terminology,  and  confusion  is 
almost  certain  to  result  from  the  use  interchangeably  of  two 
sets  of  names  for  one  set  of  ideas. 

The  chief  difficulty  of  the  method  of  limits  lies  in  acquir- 
ing a  clear  understanding  of  the  notion  of  a  limit,  and  of  its 
application  to  functions  of  one  or  more  variables.  This 
notion  cannot  be  eliminated  from  even  elementary  mathe- 
matics, and  attempts  to  evade  it  must  be  futile.  Experience 
has  shown  that  when  fairly  faced,  it  offers  no  serious  diffi- 
culty to  beginners,  and  when  once  it  has  been  grasped,  the 
development  of  our  subject  proceeds  with  ease,  security,  and 
economy  of  energy.  One  who  has  once  acquired  a  fair 
knowledge  of  the  Calculus  by  the  method  of  limits,  will 
have  no  difficulty  of  consequence  in  understanding  the  dif- 
ferential notation,  should  he  happen  to  meet  it  later  in  his 
reading. 

A  word  as  to  rigor.  The  present  may  be  styled  the  "  Age 
of  Rigor"  in  the  development  of  the  Calculus.  The  keen 
tliinkers  of  the  generation  of  mathematicians  just  passing 
away  found  much  in  the  work  of  the  older  masters  that 
needed  more  precise  formulation  and  more  strict  treatment. 
But  as  it  was  not  natural  or  easy  in  the  beginning  of  the 
subject  to  perceive  all  the  underlying  subtle  discriminations 
which  were  seen  later,  so  now  it  is  highly  inadvisable,  if  not 


vm  PREFACE 

quite  impossible,  to  present  the  subject  to  beginners  in  the 
careful  form  which  the  modern  notion  of  rigor  demands. 
Nevertheless,  an  introduction  to  the  Calculus  to-day  should 
profit  by  the  results  of  the  nineteenth  century's  labors.  '  In 
the  present  work  the  fundamental  principles  and  methods 
have  been  treated  in  as  careful  a  manner  as  seemed  consist- 
ent with  the  elementary  character  of  the  work,  and  through- 
out the  aim  has  been  to  give  a  presentation  in  harmony,  at 
least,  with  the  more  strict  treatment,  and  permitting  later 
extension  upon  the  foundation  here  laid. 

In  the  choice  of  the  exercises,  the  aim  has  been  to  exem- 
plify, to  clarify,  and  to  fix  in  mind  the  principles  which  have 
been  explained.  To  this  end  the  exercises  are  simple  in 
character,  so  that  the  application  of  the  principle  may  not 
be  obscured  by  complexity  of  computations.  The  number 
of  the  exercises  is  thought  to  be  sufficient  to  attain  the  end 
in  view,  though  not  sufficient  to  insure  the  attainment  of 
great  dexterity  in  the  handling  of  long  and  intricate  expres- 
sions. This  skill  can  be  attained  only  by  considerable  prac- 
tice after  the  principles  are  understood.  As  the  topics 
treated  are  those  usually  taken  up  in  accordance  with  well- 
established  usage,  the  teacher  who  desires  to  do  so,  can 
readily  select  supplementary  exercises  from  other  sources. 

It  is  much  more  difficult  to  secure  from  works  on  the 
physical  sciences  good  illustrative  examples  sufficiently  sim- 
ple to  be  available  for  the  present  work.  When  found,  Ihey 
usually  require  alteration  in  form  to  bring  them  into  uni- 
formity with  one  another  and  with  our  treatment  of  the 
subject.  Accordingly  the  number  of  such  examples  included 
is  perhaps  larger  than  needful  for  any  one  class,  permitting 
the  teacher  to  make  such  selection  as  he  may  deem  wise. 
The  illustrations  from  the  physical  sciences  are  usually 
independent  of  one  another,  and  any  or  all  of  them  may  be 
omitted  without  breaking  the  course  of  the  mathematical 
development. 


PREFACE  IX 

The  chapters  are  also  in  the  main  independent  of  one 
another,  at  least  to  such  an  extent  as  to  permit  whatever 
omissions  or  variations  in  the  order  of  reading  are  likely  to 
be  desired.  In  particular,  it  is  possible,  without  serious 
inconvenience,  to  take  up  first  all  the  chapters  relating  to 
the  Differential  Calculus. 

The  first  chapter  consists  of  an  introduction  to  Analytic 
Geometry,  and  contains  all  that  is  presupposed  from  this 
subject  in  the  remainder  of  the  book.  This  chapter  may 
be  omitted  by  those  who  have  already  had  a  course  in 
Analytic  Geometry. 

The  historical  notes  are  in  some  instances  based  on  exami- 
nation of  the  original  sources  by  us,  but  usually  on  the 
authority  of  Cantor.* 

It  is  hoped  that  the  work  as  here  presented  may  be  helpful 
to  students  of  several  types :  — 

To  the  student  of  mathematics  as  a  pre-view.  In  perhaps 
every  branch  of  mathematics  the  subject-matter  may  readily 
be  divided  roughly  into  fundamental  principles,  methods,  and 
results,  which  are  not  difficult  of  comprehension,  and  their 
combinations  and  generalizations,  which  may  grow  to  any 
degree  of  complexity.  It  is  usually  a  mistake  to  combine 
these  two  divisions  of  the  subject-matter  in  a  first  presenta- 
tion. When  once  the  fundamental  principles  and  results 
are  well  in  hand,  the  attention  can  be  given  entirely  to  the 
steps  by  which  these  are  combined  into  more  elaborate 
results,  and  thus,  taken  in  due  order,  the  complex  conse- 
quences offer  no  more  difficulty  than  the  simple  elements ; 
while  detailed  treatment  of  topics  whose  fundamental  prin- 
ciples have  not  been  thoroughly  digested,  entails  unnecessary 
difficulty  if  not  absolute  failure.  To  the  prospective  student 
of  mathematics  the  present  work  offers  such  a  first  general 

*  Cantor,  Vorlesungen  iiber  Geschuhte  der  Mathematik,  Bde.  II.  III. 


X  PREFACE 

survey  of  the  field  of  the  Calculus,  and,  if  desired,  of  Ana- 
lytic. Geometry  also. 

To  the  general  student  as  a  part  of  liberal  culture.  The 
reasons  for  believing  that  a  course  in  the  Calculus  should 
round  out  the  mathematical  study  of  the  general  student 
have  already  been  touched  upon.  The  earliest  stage  at 
which  work  in  mathematics  may  properly  cease  in  the 
attainment  of  a  liberal  education  has  been  well  indicated 
by  Hill  :  *  — 

"  How  far  is  mathematical  study,  then,  to  be  insisted  upon 
as  necessary  to  a  'liberal'  education?  Certainly  no  educa- 
tion can  be  called  '  liberal '  which  has  not  enabled  the  recipi- 
ent of  it  to  perceive  the  mathematical  necessity  that  runs 
through  all  natural  relations,  and  to  make  those  calculations 
which  are  needed  in  the  exact  sciences." 

To  the  student  of  natural  science^  as  giving  sufficient  of  an 
acquaintance  with  the  Calculus  to  render  certain  important 
recent  developments  in  his  domain  intelligible. 

To  the  student  of  astronow,y^  of  advanced  physics^  of  tech- 
nology, for  the  same  reasons  as  to  the  student  of  mathematics. 

Various  professional  colleagues  were  good  enough  to  look 
over  portions  of  the  proofs  and  to  give  us  valuable  sugges- 
tions which  we  have  utilized.  We  wish  to  thank  all  of 
these  gentlemen  most  heartily  for  this  assistance,  as  well  as 
the  publishers  for  facilitating  the  work  in  every  way  in  their 
power. 

J.  W.  A.  YOUNG, 
C.  E.  LINEBARGER. 

*  Hill,  The  American  College  in  Belation  to  Liberal  Education.  Inau- 
gural address  as  President  of  the  University  of  Rochester,  p.  19. 


TABLE    OF    CONTENTS 


CHAPTER   I 
THE  ELEMENTS  OF  ANALYTIC  GEOMETRY 

ARTICLE                                                                                                                                                                 .  PAGE 

1.  Graphic  representation ,1 

2.  Coordinates 7 

Exercises  I    .        . 10 

3.  The  fundamental  principle  of  Analytic  Geometry      ...  12 

Exercises  II  .        .        .        .        .                 ....  15 

4.  The  equation  of  the  circle 16 

Exercises  III 19 

5.  The  equation  of  the  parabola 20 

6.  The  equation  of  the  straight  line  through  the  origin          .         .  22 

7.  The  equation  of  any  straight  line 24 

8.  Every  equation  of  the  first  degree  represented  by  a  straight  line  27 

9.  The  intercepts 28 

Exercises  IV 29 

10.  Gay-Lussac's  Law 29 

11.  Problems  on  the  straight  line 30 

12.  Concerning  the  nature  of  a  general  equation      ....  35 

Exercises  V 36 

13.  Two  straight  lines ...  38 

Exercises  VI 41 

14.  The  equation  of  the  ellipse 42 

15.  The  form  of  the  ellipse         .         .  -         .         .         .         .44 

16.  Problems  concerning  the  ellipse  . 45 

17.  The  auxiliary  circle ;  the  directrix ;  the  eccentricity       '    .         .50 

Exercises  VII 54 

18.  The  equation  of  the  hyperbola 55 

19.  The  form  of  the  hyperbola 57 

20.  The  directrix  of  the  hyperbola    .         .   •     .         .         .         .         .57 

21.  The  equilateral  hyperbola  and  its  asymptotes     ....  59 

Exercises  VIII 62 

xi 


xu 


TABLE  OF  CONTENTS 


ART.  PAGE 

22.  Transformation  of  coordinates    .......  63 

Exercises  IX 65 

23.  Van  der  Waal's  equation ,         .  66 

24.  Polar  coordinates 69 

25.  The  equations  of  the  ellipse,  the  parabola,  and  the  hyperbola  in 

polar  coordinates .70 

26.  The  spiral  of  Archimedes 74 

27.  Concerning  imaginary  points  and  lines       .....  75 

CHAPTER   II 


CONCERNING  LIMITS 


1. 
2. 
3. 
4. 

5. 

6. 

7. 

8. 

9. 
10. 
11. 


Constants,  variables,  and  limits 
Illustrations  of  limits  . 


Definition  of  limit.     Rigorous  definition  of  limit 
Application  of  the  definition;  further  ilhistrations 
Concerning  infinity      .... 
Further  examples  of  limits  . 
The  fundamental  theorem  of  limits     . 
Propositions  concerning  limits     . 
Concerning  epsilons      .... 
Properties  of  epsilons  .... 
Proof  of  the  propositions  concerning  limits 
Exercises  X 


77 
79 
80 
82 
85 
86 
87 
90 
91 
92 
92 
95 


CHAPTER   III 

THE  FUNDAMENTAL  CONCEPTIONS   OF  THE  DIFFERENTIAL 
CALCULUS 


1.  The  underlying  principles    . 

2.  Motion  on  the  parabola 

3.  Concerning  speed  .... 

4.  The  motion  of  a  freely  falling  body     . 

5.  The  linear  expansion  of  a  rod 

6.  The  derivative 

7.  The  physical  signification  of  derivatives 

8.  The  function-concept    .... 

Exercises  XI         .... 

9.  General  rule  for  the  formation  of  derivatives 

Exercises  XII 


97 
99 
102 
102 
105 
107 
109 
110 
114 
115 
118 


TABLE  OF  CONTENTS  xiii 


CHAPTER  IV 
•     DERIVATIVES  OF   THE   SIMPLER  FUNCTIONS 

ART.  PAGK 

1.  The  derivative  of  x"     .         .         . 120 

2.  The  derivative  of  sin  a:  aiul  cos  x 121 

3.  Geometric  interpretation  of  the  sign  of  the  derivative        .         .  124 

4.  Derivatives  of  sums  and  differences     ......  126 

5.  The  derivative  of  cf(x),  c  being  a  constant         ....  128 

6.  The  derivative  of  a  constant .  128 

Exercises  XIII      .        .        . 129 

7.  The  derivative  of  a  product 130 

Exercises  XIV 132 

8.  The  derivative  of  a  quotient        .         .         .         .         .         .         .  132 

Exercises  XV 135 

9.  Logarithmic  functions           ........  136 

10.  Relations  between  logarithms  with  different  bases     .         .         .  140 

11.  Connection  between  -^  and  -r~    ......        .  142 

ax  dy 

12.  The  exponential  function .         .  143 

lo.   Illustrative  discussion  of  the  exponential  function      .         .         .  144 

14.  Inverse  trigonometric  functions 147 

Exercises  XVI 149 

15.  Differentiation  of  functions          . 150 

Exercises  XVII 154 

16.  The  derivative  of  a  power  with  any  exponent     .         .         .         .  155 

17.  Logarithmic  differentiation .  157 

18.  Summary  of  results       .........  158 

Exercises  XVIII  (Miscellaneous) 159 

19.  Continuity  and  discontinuity 160 

CHAPTER  V 

THE  FUNDAMENTAL  CONCEPTIONS  OF  THE  INTEGRAL 
CALCULUS 


1.  The  problems  of  the  integral  calculus 

2.  Integrals 

3.  The  integral  calculus  as  an  inverse  problem 

4.  The  constant  of  integration         .... 

5.  The  fundamental  formulae  of  the  integral  calculus 

6.  The  geometric  signification  of  the  constant  of  integration 

7.  The  physical  signification  of  the  constant  of  integration   . 


166 
169 
172 
174 
175 
176 
180 


XIV  TABLE  OF  CONTENTS 

CHAPTER   VI 
THE  SIMPLER  METHODS  OF  INTEGRATION 

AUT,  PAGE 

1.  Integration  of  sums  and  differences     ......  184 

Exercises  XIX »        .  185 

2.  Integration  by  the  introduction  of  new  variables        .         .         .  186 

Exercises  XX ,        .  191 

3.  Integration  by  parts     .........  191 

Exercises  XXI 195 

4.  On  special  artifices 195 

5.  Integration  by  transformation  of  the  function  to  be  integrated  .  196 

6.  Formulae  of  reduction 199 

7.  Integration  by  inspection 200 

Exercises  XXII 201 

8.  Decomposition  into  partial  fractions 203 

Exercises  XXIII .        .        .213 

9.  Summary  of  results 213 

Table  of  Integrals .        .  214 

Exercises  XXIV  (Miscellaneous) 216 

CHAPTER  VII 
SOME  APPLICATIONS  OF  THE  INTEGRAL  CALCULUS 

1.  The  attraction  of  a  rod 218 

2.  The  hypsometric  formula     . ■        .  220 

Exercises  XXV 223 

3.  Newton's  law  of  cooling ;         .  224 

4.  Concerning  the  general  method  of  all  these  applications    .         .  229 

5.  Work  done  in  the  expansion  of  a  perfect  gas  at  a  constant 

temperature 230 

6.  Work  done  in  the  expansion  of  a  highly  compressed  gas  kept 

at  constant  temperature 232 

7.  Work  done  in  the  expansion  of  a  gas  undergoing  dissociation 

at  constant  temperature 233 

8.  Maximum  average  temperature  of  a  flame  .         .         .         ,         .  236 

9.  Chemical  reactions  in  which  the  factors  are  totally  converted 

into  products 240 

10.  Reactions  in  which  the  factors  are  only  partially  converted 

into  products         .         , 242 

11.  Formation  of  lactones           . •  243 


TABLE  OF  CONTENTS  XV 

CHAPTER   VIII 
DEFINITE  INTEGRALS 

ART.  PAGE 

1.  The  quadrature  of  the  parabola .  245 

2.  Xotation  of  sums c         .         ,         .  248 

3.  The  quadrature  of  an}^  curve 249 

4.  Definite  integrals ,         .  253 

5.  The  quadrature  of  the  ellipse  and  of  the  hyperbola    .         ,         .  255 

6.  The  volume  of  a  solid 258 

7.  The  volume  of  the  sphere  and  of  the  paraboloid  of  revolution   .  259 

8.  The  mass  of  a  rod  of  varying  density 261 

9.  Some  laws  of  operation  for  definite  integrals      ....  262 

Exercises  XXVI 265 

10.  The  rectification  of  curves 267 

Exercises  XXVII          ....,.,.  268 

11.  Definite  and  indefinite  integrals .         .         .         •      .  <•         •         »  269 


CHAPTER  IX 

HIGHER  DERIVATIVES  AND  FUNCTIONS  OF  SEVERAL 
VARIABLES 

1.  Definition  of  higher  derivatives  . 272 

2.  The  higher  derivatives  of  the  simplest  functions         .         .         .  273 

Exercises  XXVIII 275 

3.  Geometric  meaning  of  the  second  derivative       ....  276 

4.  Physical  interpretation  of  the  second  derivative  .         .         .  278 

5.  Oscillatory  motion 280 

6.  The  velocity  acquired  by  a  body  falling  toward  the  earth  from 

a  great  distance 282 

7.  Partial  derivatives 284 

8.  Higher  partial  derivatives 288 

Exercises  XXIX 290, 

9.  Differentiation  of  a  function  of  two  or  more  functions  of  a 

single  independent  variable 291 

Exercises  XXX 295 

10.  Differentiation  of  implicit  functions    .         .         .         .         .        .  295 

Exercises  XXXI 298 

11.  Homogeneous  functions 298 

Exercises  XXXII  ...,.,..  299 


XVI  TABLE  OF  CONTENTS 

ART.  PAGE 

12.  Euler's  theorem  of  homogeneous  functions         o         .         .         .  299 

Exercises  XXXIII ,        .300 

13.  The  focal  properties  of  the  parabola 301 

14.  The  focal  properties  of  the  ellipse 303 

15.  The  asymptotes  of  the  hyperbola         ......  306 


CHAPTER  X 

INFINITE  SERIES 

1.  Definition .310 

2.  The  sum  of  infinite  series .  311 

3.  The  geometric  series 313 

4.  General  theorems  on  the  convergence  of  series.     Series  with 

alternating  signs  . 314 

Exercises  XXXIV 317 

5.  Series  with  varying  signs 318 

6.  Series  whose  signs  are  all  positive 320 

7.  Rapidity  of  convergency 322 

8.  Application  to  the  series  f or  e      .         .         .         .         .                 .  323 

Exercises  XXXV          .        .        .        .        .        .        .        .  324 

9.  Maclaurin's  Theorem ,  325 

10.  The  series  for  e=*,  sin  x  and  cos  x 328 

Exercises  XXXVI 332 

11.  The  series  for  tan  a:       .         .         .         .         .         .         .         ,         .  332 

12.  Taylor's  Theorem .         .  334 

13.  The  logarithmic  series 337 

14.  The  binomial  theorem 340 

Exercises  XXXVII       .......  342 

15.  Integration  by  series    .........  342 

16.  Table  of  series      .........  346 

17.  Indeterminate  forms 347 

18.  Illustrative  examples  of  the  determination  of  the  limits  of  in- 

determinate forms 350 

19.  Types  of  indeterminate  forms 355 

Exercises  XXXVIII     .........  356 

20.  Calculation  with  small  quantities        ......  357 

21.  Reduction  with  barometric  readings  to  0°  C.      .        .        .        .  358 

22.  Simplified  hypsometric  formula  ...        ...         .  359 


TABLE  OF  CONTENTS  xvii 


CHAPTER   XI 
MAXIMA  AND  MINIMA 

ART.  PAGE 

1.  Conditions  for  a  maximum  or  minimum     .         .         .,         ,         ,  361 

2.  Points  of  inflexion  of  curves         ..,...,  364 

Exercises  XXXIX 367 

3.  Exceptional  cases ;  general  theory       ......  367 

4.  Collected  criteria  concerning  forms  of  curves      ,         .         .         .  368 

5.  Examples  of  maxima  and  minima       ......  369 

6.  Minimum  of  intensity  of  heat      .......  374 

7.  The  law  of  reflection 376 

8.  The  law  of  refraction ,         .        ,       \  378 

Exercises  XL         . 380 

9.  Estimation  of  errors ,         .        .         .  383 


CHAPTER  XII 

DIFFERENTIATION  AND  INTEGRATION  OF  FUNCTIONS  FOUND 
EMPIRICALLY 

1.  Differentiation      ..........    389 

2.  Integration 395 

APPENDIX 

Collection  of  Formulae        .        .        .        .        .        .        .        ,    401 

Index .    405 


CALCULUS 

CHAPTER   I 
THE  ELEMENTS  OF  ANALYTIC  GEOMETRY 

Art.  1.  Graphic  representation.  During  the  last  few 
decades,  the  Graphic  Methods  have  developed  more  and 
more  into  general  and  useful  aids  in  investigation.  They 
are  employed  to  great  advantage  in  the  physical  as  well 
as  in  the  descriptive  sciences,  such  as  Geography,  Meteor- 
ology, Physiology,  Sociology,  etc.  ;  and  are  applicable,  in 
short,  wherever  laws  and  rules  are  considered  in  connection 
with  numbers.  The  peculiar  value  Avhich  these  methods 
possess  lies  in  the  substitution  of  geometric  figures  for 
numerical  tables,  the  relations  of  the  numbers  being  thus 
made  directly  apparent  to  the  eye.  A  few  exaniples  will 
suffice  to  show  the  importance  and  applicability  of  the 
graphic  method. 

I.  As  a  first  illustration  we  reproduce*  a  table  and 
diagram  giving  the  value  of  various  elements  of  growth  of 
the  United  States  at  the  times  indicated.  Much  that  must 
be  slowly  gleaned  from  the  table  of  statistics  alone  is  told  by 
the  diagram  at  a  glance.  From  it  we  can  answer  at  once 
such  questions  as  :  When  were  the  carrying  trades  in  Ameri- 

*  W.  J.  McGee,  The  National  Geographic  Magazine^  September,  1898. 

1 


2   • 


VA'LCULUS 


[Ch.  I. 


can  and  foreign  bottoms  equal  ?     When  did  the  density  of 
population  increase  ?     When  decrease  ?     etc.,  etc. 

II.    What  is  known  as  Boyle's  Law  states  the  relation  in 
which  the  pressure  and  the  volume  of  a  gas  stand  when  all 


Elements  of  Growth 


Area  square  miles  .        .  827,S44 

Total  population     .        .       3,929,214 

Population-density         .  4. 75 

Wealth    . 

Wealth  per  capita  , 

Kailway  mileage 

Carrying  trade,  foreign  bottoms 

Carrying  trade,  American  bottoms 


1800 


827,844 

5,308,483 

6.41 


1810 


1,999,775 

7,239,881 
3.62 


1820 


1,999,775 
9,633,822 

4.82 


$14,358,235 
$113,201,462 


18:30 


2,059,043 

12,866,020 

6.25 


$14,447,970 

$129,918,458 


Elements  of  Growth 


Area  square  miles 

Total  population  . 

Population-density 

Wealth  .... 

Wealth  per  capita 

Eailway  mileage  . 

Carrying  trade,  foreign  bottoms 

Carrying  trade,  American  bottoms 


2,059,04:3 
17,069,453 

8.29 


2,818 

$40,802,856 

$198,424,609 


1850 


2,980,959 

23,191,876 

7.78 

$7,136,000,000 

$308 

9,021 

$90,7W,954 

$239,272,084 


1860 


3,025,600 

31,443,321 

10.39 

$16,160,000,000 

$514 

30,626 

$255,040,793 

$507,^7,757 


1870 


3,556,600 

38,558,371 

10.84 

$30,069,000,000 

$780- 

52,922 

$638,927,488 

$352,969,401 


Elements  of  Growth 


Area  square  miles 

Total  population  . 

Populatio  n  -den  sity 

Wealth  . 

Wealth  per  capita 

Kailway  mileage  . 

Carrying  trade,  foreign  bottoms 

Carrying  trade,  American  bottoms 


1880 


3,556,600 

50,155,783 

14.10 

$43,&42,000,000 

$870 

93,296 

$1,224,265,434 

$258,346,577 


1890 


3,556,600 

62,622,250 

17.61 

$65,037,091,197 

$1,036 

166,691 

$1,371,116,744 

$202,451,086 


1898  a 


8,556,600 

71,000,000 

20.00 


190,000 

$1,600,000,000 

$190,000,000 


1898  & 


3,681,236 

79,000,000 

21.46 


other  properties  are  kfept  constant.  A  certain  mass  of  gas 
is  at  one  time  under  the  pressure  p^  and,  at  another  time, 
under  the  pressure  p^;    ii   v  and  v^  are  the  volumes  cor- 


1.] 


The  elements  of  analytic  geometry 


responding  to  these  pressures,  then  Boyle's  Law  states  that 
V  and  v^  are  inversely  proportional  to  p  and  ^^ ;  that  is,  we 
have  the  proportion, 

or  the  equation, 

(1)  pv=p^v^. 


Fig.  1. 


The  graphic  representation  of  Boyle's  Law  is  obtained  by 
making  use  of  this  equation  in  the  following  way.  First  of 
all,  a  series  of  corresponding  values  of  pressure  and  volume 
is  determined.  If  it  be  assumed  that  Pi  =  1  and  the  corre- 
sponding ^1  =  1,  then  equation  (1)  becomes  pv  =  l^  and  the 


[Ch.  I. 


following  table  gives  the  con 
li  p  equals, 

0.1     0.2     0.5     1 


CALCULUS 

•esponding  values  of  p  and  v. 


5  etc. 


the  corresponding  values  of  v  are,  respectively, 

10      5        2        1      0.5     0.25     0.2  etc. 

We  now  draw  (Fig.  2)  any  horizontal  line  whatever  and 
measure  off  on  it  from  a  point  such  as  0,  distances  equal  to 
the  values  of  p  respectively,  that  is, 
equal  to  0.1,  0.2,  0.5,  1,  2,  4,  etc.;  at 
these  points  of  division  we  erect  perpen- 
diculars whose  lengths  shall  be  equal 
to  the  corresponding  values  of  v^  or  to 
10,  5,  2,  1,  0.5,  0.25,  etc.^  If  the  ex- 
tremities of  these  perpendiculars  be  con- 
nected by  a  curved  line,  the  curve  thus 
plotted  is  the  graphic  representation  of 
Boyle's  Law.  It  is  at  once  apparent 
that  it  is  necessary  to  determine  the 
position  of  quite  a  large  number  of 
points  in  order  to  obtain  the  course  of 
the  curve.* 

III.  The  air  in  a  soap-bubble  is  compressed  more  than  the 
external  air,  as  is  evident  from  its  diminishing  in  size  Avlien 
the  stem  of  the  pipe  used  to  blow  it  is  left  open.  The  pres- 
sure of  the  confined  air  and  the  diameter  of  the  bubbles  are 
given  in  the  following  table  : 


1 

iu 

u 

- 

8 

7 

5 

4 

i 

\ 

V 

> 

1 

s 

^ 

"^ 

-* 

*~ 

18       3 
Fig.  2. 


*  It  is  convenient  to  employ  tlie  so-called  "coordinate  or  cross-section 
paper,"  (i.e.  paper  divided  into  small  squares  by  lines  ruled  at  right  angles,) 
for  drawing  the  graphic  representation  of  a  law. 


■ 


1.] 


THE  ELEMENTS   OF  ANALYTIC  GEOMETRY 


d 

Diameter  of  Bubble  in 

Millimeters 

p 

Pressure  of  Internal 
Air* 

dp 

7.55 

3.00 

22.65 

10.37 

2.17 

22.50 

10.55 

2.13 

22.47 

23.35 

0.98 

22.88 

27.58 

0.83 

22.89 

46.60 

0.48 

22.37 

Mean  =  22.63 

An  inspection  of  the  columns  of  numbers  shoAvs  that  as 
d  increases,  p  decreases,  and  that  their  product  is  nearly 
constant.  Accordingly  we  may  put  dp  =  constant ;  the 
diameter  of  a  soap-bubble  is  inversely  proportional  to 
the  pressure  of  the  air  contained  in  it.  Constructing 
the  graphic  representation,  we  obtain  a  curve  similar  in 
shape  to  that  for  Boyle's  Law  as  given  above. 

IV.  If  a  solid  be  brought  into  contact  with  a  liquid,  it 
often  happens  that  the  solid  dissolves  in  the  liquid,  forming 
a  solution.  There  is  a  limit  to  the  amount  of  a  solid  that 
will  dissolve,  and  when  this  limit  is  reached  the  solution  is 
said  to  be  saturated  with  the  solid ;  the  percentage  by  weight 
of  the  solid  contained  in  the  saturated  solution  is  termed 
the  solubility  f  of  the  solid  in  that  liquid. 

Solubility  generally  changes  with  a  rise  or  fall  of  tempera- 
ture, being  different  at  different  temperatures.  The  solu- 
bility of  cane  sugar  in  water  has  been  determined  to  be : 

*  The  unit  for  the  pressure  of  the  internal  air  is  the  pressure  of  air  neces- 
sary to  support  or  counterbalance  a  vertical  column  of  water  one  millimeter 
in  height. 

t  Of  course,  solubility  may  be  expressed  in  other  ways,  as  parts  of  liquid 
required  to  dissolve  one  part  of  solid,  etc. 


6  CALCULUS  [Ch.  L 

Temperature 0°       10°     20°    30°    40°    50° 

Percentage  of  sugar  dissolved  .     .     .     65.0  65.6  67.0  69.8  75.8  82.7 

If  on  any  horizontal  line  we  measure  off  distances  equal 
to  0,  10,  20,  30,  40,  50  units  on  any  chosen  scale,  at  the 
points  of  division  erect  perpendiculars  equal  to  65.0,  65.6, 
67.0,  69.8,  75.8,  82.7  units  (on  a  scale  which  may  or  may  not 
be  the  same  as  that  employed  in  marking  the  temperatures), 
and  then  connect  with  straight  lines  the  ends  of  the  per- 
pendiculars, we  find  we  have  constructed  a  broken  line 
which  resembles  a  curve.  If  the  solubility  had  been  deter- 
mined at  any  intermediate  temperatures,  we  should  have  a 
still  more  frequently  broken  line,  and  it  is  easily  seen  that 
the  more  values  we  determine  for  the  solubility  at  differ- 
ent temperatures,  the  more  closely  does  our  broken  line 
approach  to  a  continuously  curved  line.  When  a  sufficient 
number  of  pairs  of  values  have  been  determined,  we  sketch  a 
continuous  curve  passing  through  all  the  points.  Such  a 
curve  is  termed  the  solubility  curve  of  sugar  in  water. 

The  practical  value  of  such  curves  may  be  illustrated  by 
the  following  considerations  :  Suppose  we  wish  to  know 
the  solubility  of  sugar  at  45°,  a  temperature  not  given  in 
the  table.  F'irst  we  sketch  the  curve  through  the  points 
determined  by  the  values  of  the  table  ;  then,  at  the  dis- 
tance 45  on  the  horizontal  line,  we  erect  a  perpendicular 
touching  the  curve  ;  the  length  of  this  perpendicular  is  the 
solubility  sought.  Moreover,  the  curve  shoAvs  clearly,  and 
at  a  glance,  the  general  effect  of  temperature  upon  the  solu- 
bility of  sugar.  At  the  lower  temperatures  the  solubility 
remains  nearly  constant,  but  at  the  higher  temperatures  it 
increases  more  and  more  rapidly. 

The  method  of  sketching  curves  just  described  is  rather 


1-2.]  THE  ELEMENTS   OF  ANALYTIC  GEOMETRY  7 

tedious  and  roundabout  when  an  accurate  figure  is  desired, 
since  a  large  number  of  pairs  of  corresponding  values  are 
required.  Unless  the  figure  is  accurately  constructed,  it 
is  more  likely  to  be  misleading  than  to  be  of  value.  There 
is,  however,  a  second  method,  which  is  much  simpler,  and  is 
derived  from  the  results  of  Analytic  Geometr}^  When 
an  equation  is  known  which  connects  the  corresponding 
numerical  values  of  two  related  quantities  as,  for  example, 
the  pressure  and  volume  of  a  gas,  or  the  pressure  and 
diameter  of  a  bubble.  Analytic  Cieometry  teaches  at  once 
and  with  complete  generality  the  properties  of  the  graphic 
representation  of  the  relationship^  which  could  he  learned 
empirically  only  through  a  tedious  operation.  This  is  true 
of  every  scientific  law  which  can  be  formulated  as  an  equa- 
tion between  two  related  quantities. 

AiiT.  2.  Coordinates.  Analytic  Geometry  is  based  upon 
the  same  fundamental  idea  as  the  method  of  graphic  repre- 
sentation, viz.  the  artifice  of  representing  pairs  of  numbers 
geometrically  by  means  of  points.* 

We  draw  in  a  plane  two  straight  lines  of  indefinite  length 
(Fig.  3),  whicli  may  make  any  angle  whatever  with  each 
other.  Their  point  of  intersection  is  designated  by  (9,  and 
the  lines  themselves  by  X'OX  and  F' OY.  We  take  in 
the  plane  of  the  figure  any  point,  as  P,  and  draw  through 


*  R^n^  Descartes  (1596-1650)  was  the  first  to  make  this  artifice  the  basis 
of  a  systematic  method  of  treating  geometric  problems.  Under  the  simple 
title  of  Geometrie,  Descartes  (Lat.  Cm'tesius)  published  in  1687  a  little 
volume  which  was  destined  to  introduce  a  new  epoch  in  the  study  of  geom- 
etry. Analytic  Geometry,  when  treated  according  to  the  methods  of 
Descartes,  is  frequently  styled  Cartesian  Geometry.  Descartes  was  a  philos- 
opher as  well  as  a  mathematician,  traveled  much,  and  led  a  varied  and 
eventful  life. 


8  CALCULUS  [Ch.  I. 

it  lines  parallel  to  the  straight  lines  OX  and    OY ;    these 
lines  cut  off  the  distances  OQ  and  OR^  which,  in  this  case, 
we  suppose  equal  to  7  and  5  units,  re- 
spectively.    We  term  the  distance  OQ 
the  abscissa  of   the   point  P,  and  the 
distance  OR  its  ordinate,  and  usually 
^     denote  these  distances  by  x  and  «/,  re- 
^  spectively;   more   briefly,  we   say  that 

for  the  point  P, 

Fig.  3.  7  -,  r 

X  =  i  and  y  =  D. 

When  it  is  not  necessary  to  distinguish  the  abscissa  and 
ordinate,  we  call  them  jointly  the  coordinates  of  the  point  P. 

A  similar  construction  can  be  made  for  any  other  point 

in  the  plane.     We  thus  obtain  for  every  point  a  definite 

abscissa  and  a  definite  ordinate,  or,  as  we  may  also  say,  a 

definite  pair  of  coordinates  expressed  in  numbers,  such  as 

7  and  5.     Conversely,   if  we  wish  to  locate  the  point  P^ 

(Fig.   4),    which   corresponds   to   the 

numbers    2.5    and    3.5,   we    have    to 

measure  off  on  the  straight  line  OX 

a  distance 

OQ^  =  2.5, 

and  on  01^  a  distance 

OR'  =  3.5, 

and  draw  parallel  lines  through   the 
points  Q'  and  R' ;  the  point  of  intersection  of   these   par- 
allels is  P'. 

Since  we  can  lay  off  distances  on  the  lines  XX'  on  either 
side  of  the  point  0,  and  likewise  on  YY'^  it  might  appear 
that  we  obtain  not  one,  but  four  points,  as  P',  P",  P'", 


2.]  THE  ELEMENTS   OF  ANALYTIC  GEOMETRY  9 

P"" .  We  avoid  this,  however,  by  establishing  the  rule  that 
such  coordifiates  as  are  measured  in  opposite  directions  from 
0  shall  he  given  opposite  signs.  Regarding,  as  heretofore, 
OQ^  and  OR'  as  positive,  we  must  regard  OQ''  and  OR"  as 
representing  negative  coordinates  ;  the  points  P\  P'\  P''\ 
P""  thus  represent  four  different  pairs  of  numbers,  viz.^ 

+  2.5,4-3.5;    -2.5,  +3.5;    -2.5,-3.5-;    +2.5,-3.5. 

The  relationship  between  the  points  of  the  plane  and  the 
pairs  of  numbers  is  accordingly  such  that  a  pair  of  numbers 
corresponds  to  every  point  of  the  plane.,  and  vice  versa^  a  point 
of  the  plane  corresponds  to  every  pair  of  numbers.^  It  is 
therefore  customary  to  speak  of  a  pair  of  numbers  as  a 
"point."  Thus  we  say,  "the  point  2,  5,"  rather  than  "the 
point  whose  abscissa  is  2  and  whose  ordinate  is  5." 

The  two  original  straight  lines  are  termed  the  axes  of 
coordinates,  the  line  X'  OX  being  the  axis  of  abscissae,  and 
Y' OY  the  axis  of  ordinates.  Each  has  a  positive  and  a 
negative  half.  They  divide  the  plane  into  four  parts,  which 
are  called  quadrants,  and  are  numbered  as  in  trigonometry, 


II 


111 


I 


IV 


The  point  0,  the  point  of  intersection  of  the  axes  of 
coordinates,  is  called  the  origin,  and  the  angle  YOX  the 
coordinate  angle;  if  it  be  a  right  angle,  as  is  found  to  be 
most  convenient  in  practice,  the  coordinates  are  called 
rectangular  coordinates. 


*  The  use  of  latitude  and  lon2;itude  to  determine  the  location  of  a  place 
upon  the  earth's  surface,  or  on  a  map,  is  based  upon  the  same  idea,  as  is 
also  (in  a  rude  way)  the  specification  of  a  house  by  its  street  and  number. 


10  CALCULUS  [Cn.  I 

Inasmuch  as  the  coordinates  of  the  pomt  P  (Fig.  3)  are 
simply  the  numbers  which  give  the  length  oi  OQ  and  OB^ 
it  follows  that  the  distances  PR  and  PQ  are  equal  respec- 
tively to  the  abscissa  and  the  ordinate  of  P.  It  is,  accord- 
ingly, sufficient  to  draw  one  of  the  parallel  lines  from  P  in 
order  to  get  the  coordinates  of  P. 

It  is  customary  to  give  tirst  the  value  of  the  distances 
parallel  to  the  axis  of  abscissae,  as  we  have  done  above. 
The  coordinates  of  the  point  are  usually  inclosed  in  paren- 
theses, as  ^'the  point  (4,  6),"  and  the  axes  are  often  called 
the  jr-axis  and  the  /-axis,  respectively. 

It  is  easily  seen  that  the  axis  of  abscissse  contains  all  the 
points  whose  ordinates  are  equal  to  zero ;  also  that  the  axis 
of  ordinates  is  the  locus  of  all  points  having  ahscissce  equal  to 
zero.  Finally,  the  origin  is  that  point  whose  coordinates  are 
both  equal  to  zero,,  corresponding  to  the  pair  of  numbers 
(0,  0). 

When  we  represent  graphically  a  point  whose  coordinates 
are  given,  we  are  said  to  construct  or  to  plot  the  point. 
Similarly,  any  polygon  or  curve  may  be  plotted  when  we 
know  the  coordinates  of  enough  of  its  points  to  determine 
it  completely.  Unless  otherwise  specified,,  the  coordinate  sys- 
tem is  hereafter  always  supposed  to  he  rectangular, 

EXERCISES    I 
1.    Plot  the  following  points  : 


(i-)    (2,3); 

(vi.)  (-1,-1); 

(xi.)   (-2,0); 

(ii.)    (4,G); 

(vii.)    (5,  -  9)  ; 

(xii.)    (-1,1); 

(iii.)    (-1,7); 

(viii.)    (0,  -  2)  ; 

(xiii.)    (-3,0); 

(iv.)    (2,  -3); 

(ix.)    (0,3); 

(xiv.)    (2,  -  7). 

(v.)    (5,  -  35)  ; 

(X.)    (0,0); 

2.]  THE  ELEMENTS   OF  ANALYTIC  GEOMETRY  11 

2.  Plot  the  straight  lines  passing  through  the  following  pairs  of  points  : 

(i.)  (8,0),  (2,3);  (vii.)  (2,  2),  (1,1); 

(ii.)  (-  2,  6),  (-  5,  -  4)  ;  (viii.)  (0,  0),  (-  3,  0)  ; 

(iii.)  (  -  3,  -  2),  (3,  2)  ;  (ix.)  (3,  -  2),  (3,  -  5) ; 

(iv.)  (0,  0),  (2,  -  1)  ;  (X.)  (  -  1,  -  2),  (3,  -  2)  ; 

(V.)  (  -  2,  5),  (3,  5)  ;  (xi.)  (2,  -  2),  (  -  2,  2). 

(vi.)  (-1,  -1),  (1,-1); 

3.  Plot  the  quadrilaterals  whose  vertices  are  the  following  sets  of 
points : 

(i.)    (2,1),  (5,2),  (6,4),  (1,5); 

(ii.)  (3,  -5),  (-4,  -2),  (0,0),  (2,3); 

(iii.)  (0,4),  (5,3),  (3,0),  (0,0); 

(iv.)  (2,  -1),(5,  -1),  (5,  -2),  (2,  -2); 

(V.)  (5,0),  (5..  4),  (-5,4),  (-5,0); 

(vi.)  (1,1),(2,4),  (-1,2),  (-2,  -1); 

(vii.)  (2,  -  1),  (5,  -  4),  (3,  -  6),  (0,  -  3). 

4.  By  inspection  of  the  coordinates,  tell  in  what  quadrants  the  tri- 
angles lie  (wholly  or  in  part)  whose  vertices  are  the  following  sets  of 
points : 

(i.)    (1,2),  (3,1),  (4,1); 

(ii.)  (-1,5),  (-2,1),  (-4,3); 

(iii.)  (2,  -  1),  (2,  -  2),  (3,  0)  ; 

(iv.)  (1,1),  (-1,2),  (-2,1); 

(v.)  (-1,  -2),  (-4,  -1),  (-1,  -1); 

(vi.)  (-1,3),  (-2,  2),  (-4,  -1). 

5.  (i.)  What  is  the  ordinate  of  any  point  which  lies  on  a  straight  line 
parallel  to  the  x-axis  and  at  the  distance  4  above  it  ? 

(ii.)  AVhat  is  the  abscissa  of  any  point  which  lies  on  a  straight  line 
parallel  to  the  ?/-axis  and  at  the  distance  d  to  the  left  of  it  ? 

(iii.)  What  is  the  abscissa  of  any  point  in  the  straight  line  perpen- 
dicular to  the  X-axis  and  intersecting  it  at  the  distance  c  from  the 
origin  ? 

(iv.)  What  relation  exists  between  the  ordinate  and  the  abscissa  of 
any  point  on  the  straight  line  which  passes  through  the  origin  and 
bisects  (a)  the  first  and  the  third  quadrant?  (b)  the  second  and  the 
fourth  quadrant? 


12 


CALCULUS 


[Ch.  I. 


Q 


Fig.  5. 


Art.  3.  The  fundamental  principle  of  Analytic  Geometry. 
The  result  of  exercise  5,  iv.  (a),  immediately  preceding,  may 
be  stated  thus :    Throughout  the  line  AB  (Fig.  5),  x  =  y^ 

and  for  all  points  not  in  AB^  x  is 
unequal  to  y  (for  x  and  y  are 
the  distances  of  any  point  from 
the  axes,  and  all  points  without 
the  bisector  are  unequally  dis- 
^  tant,  i.e.  for  all  such  points  x^y^. 
That  is,  the  relation  x=y  v^  char- 
acteristic of  this  straight  line  and 
no  other,  and  x  =  y  may  hence  be 
called  the  equation  of  this  line. 
The  method  we  have  employed  in 
this  simple  case  holds  generally  :  if  any  curve  is  given,  and 
if  we  can  succeed  in  finding  a  relation  between  x  and  y  which 
holds  for  the  coordinates  of  every  point  of  the  curve,  and 
which  does  not  hold  for  even  a  single  point  outside  of  the 
curve,  the  relation  so  found  is  characteristic  of  the  curve,  and 
if  the  relation  can  be  expressed  by  an  equation,  the  latter 
may  be  called  the  equation  of  the  curve. 

Conversely,  if  there  is  given  an  equation  expressing  a  rela- 
tion between  the  coordinates  {x  and  ?/)  of  a  point,  the  question 
arises :  does  there  exist  a  curve  such  that  the  given  relation 
exists  between  the  coordinates  of  every  point  of  it,  and  does 
not  exist  between  the  coordinates  of  any  other  point  w^hatso- 
ever?  This  question  can  usually  be  answered  in  the  affirma- 
tive, as  we  proceed  to  illustrate  in  a  few  simple  examples. 
Suppose  that  we  have  given  an  equation  between  x  and  y, 
which  we  assume  to  be  of  the  simplest  possible  form,  as 


(1) 


a;  +  y  =  4. 


3.] 


THE^ ELEMENTS   OF  ANALYTIC  GEOMETRY 


13 


In  the  equations  of  elementary  mathematics,  the  problem  is 
to  determine  the  values  of  the  "  unknown  quantities,"  which 
will  satisfy  the  equation ;  each  problem  usually  has  but 
a  finite  number  of  definite  solutions.  Now,  however,  the 
state  of  affairs  is  quite  different ;  for  there  are  countless 
pairs  of  numbers  which,  when  introduced  into  our  equation 
for  X  and  ?/,  will  satisfy  the  equation.  Such  pairs  of  num- 
bers are,  for  example  : 


x=5 

»=4 

x=Z 

x=1 

y=-l 

y  =  0 

y  =  \ 

y=2 

x=l 

x=-l 

x=-\.b 

x=~% 

2/=3 

y  =  b 

y=5.5 

y=Q 

Whatever  number  we  may  take  for  a;,  we  can  always  obtain 
from   equation  (1)  a  corresponding  value  for  y  such   that 
the  pair  of  values  so  determined  will     * 
satisfy   the   equation.     For   each    of 
these   pairs  of   numbers,   a   point  of 
the  plane  can  be  determined  whose 
coordinates  are  the  numbers  taken  ; 
in   this   way   we   obtain   (Fig.   6)    a 
boundless    number    of    points,  all  of 
which  lie  upon  a  definite  geometric 
curve  that  in  this  case  seems,  in  the 

figure,  to  be  a  straight  line;   and  it  " 

will  be  shown  further  on  that  this  is  actually  the  case. 
As  a  second  example  we  take  the  equation, 


-!s         y 

S 

s 

s 

^ 

s 

5^     i 

\ 

1         5 

s 

s 

s^ 

_S^  '■ 

0                -S^ 

-4-                     5^ 

^ 

Fig.  6. 


(2) 


y^=2x; 


14 


CALCULUS 


[Ch.  I. 


by  assigning  to  x  the  values  0,  1,  2,  3  .   .   . ,  we  obtain  the 
following  pairs  of  numbers  which  satisfy  the  equation : 

x=^        x=l  x=2  x=^ 

2/  =  0       ?/=±V2=±1.4.  .      ^=±2       ^=±V6  =  ±2.45.. 

X  =  4:  x=5 

y=±V8=±2.8  .  .  y=±VlO=±3.2  .  . 

For  every  value  of  x  there  are  two  diiferent  values  of  ^; 

thus  for  o 

ic  =  2, 

and  y  =  —  2, 

so  that  both  the  point  P'  corresponding  to  the  pair  of  numbers 

x  =  2 
and  ^  =  2, 

and  the  point  P'^  to  which  the  numbers 

x  =  2 
and  y  =  —  2' 

belong,  have  coordinates  which  satisfy  equation  (2).  Any 
number  of  such  points  can  be  found,  and  all  of  them  lie  upon 
a  definite  geometric  curve  (Fig.  7), 
which  is  called  (Art.  5)  a  parabola. 

Similar  considerations  can  be  applied 
to  every  other  equation  between  x  and 
y.  The  countless  pairs  of  numbers 
that  can  satisfy  any  equation  always 
correspond  to  countless  points,  all 
usually  lying  upon  a  certain  curve. 
Fig.  7.  The    relation   between    the    equation 


~ 

... 

' 

^ 

«^ 

" 

^ 

^ 

^ 

/" 

f 

X 

- 

(E 

I 

' 

) 

V 

s 

s 

S 

1 

y 

^ 

•> 

f*-i 

_ 

_j 

_j 

_ 

_ 

_ 

_ 

3.]  THE  ELEMENTS   OF  ANALYTIC  GEOMETRY  15 

and  the  curve  may  be  expressed  by  stating  that  all  points 
whose  coordinates  satisfy  the  equation  in  question  lie  upon 
the  curve  and  (as  will  appear  more  fully  later)  the  coordi- 
nates of  every  point  upon  the  curve,  used  as  values  of  x  and 
y,  will  satisfy  the  equation.  We  often  express  this  relation 
still  more  briefly  by  saying  :  the  given  equation  is  the  equa- 
tion of  the  curve.  The  curve  corresponding  to  an  equation 
is  often  called  its  graph,  and  also  its  locus.  The  curve 
regarded  as  the  graph  of  an  equation,  may  be  considered 
the  picture  of  the  equation.  Properties  of  the  graph  may 
be  discovered  by  studying  the  equation,  and  vice  versa. 
To  do  this  "is  the  fundamental  purpose  of  Analytic  Geom- 
etry. 

A  wide  perspective  now  opens  up  before  us ;  we  see  what 
must  first  be  done  in  Analytic  Geometry.  Two  problems 
present  themselves  at  once  :  (1)  to  find  the  curve  belonging 
to  any  given  equation,  and  (2)  to  ascertain  what  is  the 
equation  of  any  given  curve.  It  will  be  sufficient  for  our 
purposes  to  consider  only  the  simplest  cases  of  both  these 
problems.  Those  of  the  second  kind  which  we  shall  need 
will  be  treated  in  the  sequel.  We  add  a  set  of  exercises 
containing  a  few  simple  examples  of  the  first  kind. 

EXERCISES   II 

Construct  the  graphs  of  the  following  equations  : 


1. 

x  =  2y. 

6. 

x'^  =  4.y. 

11. 

dx-4:  =  2y 

2. 

x  +  y  =  0. 

7. 

7/  +  22;  +  4nr0. 

12. 

x'^  +  y'^  =  25. 

3. 

2x-y  =  0. 

8. 

^2  +  2/2  ^  36. 

13. 

a:2=5?/-2. 

4. 

y  =  x-\-2. 

9. 

4a:2  =  9?/2. 

14. 

2/  =  6. 

5. 

x  =  y^ 

10. 

x-y  -2  =  0. 

16  CALCULUS  [Ch.  I. 

Art.  4.    The  equation  of  the  circle.      G-iven   a   circle   of 
radius  r ;   to  find  its  equation. 

Let  us  take  the  axes  of  coordinates  so  that  they  shall 
intersect  at  tlie  center  of  the  circle  and  form  a  right  angle 

(Fig.  8).  The  coordinates  of  any 
point  of  the  circle  as  P^  are  OQ-^ 
and  Pjd,  which  we  call  x^  and  y^. 
In  the  right  triangle  OP^Q^, 


OQ^^P^Q^=^OP^\ 

or,  if   a7j,  2/^,  r,  be  substituted  for 


Fig.  8. 


(1) 


^\  +  y\  =  ^^• 


If  Pg  ^^®  ^  second  point  of  the  circle,  and  OQ^  and  P2Q21 
or,  X2  and  t/^  ^^  i^^  coordinates,  in  a  perfectly  analogous 
manner,  we  obtain  from  the  right  triangle  OP^Q^ 


OQ,'  +  Q,P,'  =  OP,\ 


or  (2) 


+  ^2^  =  ^^• 


In  the  same  way  it  follows  that,  for  any  third  point  as  Pg 
with  the  coordinates  OQ^  and  P^Q^,  or  x^*  and  3/3, 


or  (3) 


0§/  +  P3§/  =  OP3^ 
^3^  +  ^3^  =  ^3^- 


*  Although  in  the  figure  Xs,  the  value  of  the  abscissa,  is  a  negative  number 
according  to  Art.  2,  yet  x-^"^  being  the  square  of  a  negative  quantity  is  posi- 
tive, and  is  therefore  also  the  square  of  the  length  of  the  side  0^3  of  the 
triangle. 


4.]  THE  ELEMENTS   OF  ANALYTIC  GEOMETRY  17 

Similar  equations  can  be  derived  for  any  point  of  the 
circle  ;  and  we  see  at  once  that  the  equation  of  the  circle 
itself  can  be  written  simply  : 


A,ix  .  2,22 

(4)  ic''  +  y  =  r^. 


an  equation  in  a;  and  ?/,  which  is  satisfied  when  (and  only 
when)  the  coordinates  of  any  point  whatever  of  the  circle 
are  substituted  for  x  and  y.  This  equation  is  satisfied  by 
the  ])airs  of  coordinates  (x-^^  ^^),  (x^^  y^^  (x^^  y^  ...  in  the 
samr  way  as  the  equations  in  Art.  3  were  satisfied  by  the 
pairs  of  numbers  there  given. 

A¥hy  does  there  exist  a  single  equation  which  is  satisfied 
by  the  coordinates  of  all  the  points  of  the  circle  ?  The 
reason  is  to  be  found  in  the  fact  that  the  relation  between 
all  the  points  of  the  circle  and  its  center  is  governed  by 
one  and  the  same  law.  Every  point  in  the  circumference 
is  equally  distant  from  the  center ;  what  is  true  for  the 
point  Pj  and  its  coordinates  x^  and  y^  is  also  true  of  the 
coordinates  of  P^  and  P^ ;  it  is  even  a  matter  of  complete 
indifference  wdiich  points  we  designate  as  P^,  P^^  P^.  In 
other  words,  all  that  we  have  to  do  in  order  to  derive  the 
equation  which  will  be  satisfied  by  the  coordinates  of  all  the 
points  of  the  circle  is  to^bbtain  the  equation  for  any  one  of 
its  points  arbitrarily  chosen  ;  and  this  one  point  is  any  point, 
hence  every  point.  Care  must  be  exercised  that  the  point 
chosen  has  no  special  or  unusually  simple  position. 

This  principle  is  of  great  importance  ;  from  now  on  we 
shall  make  continual  use  of  it  in  obtaining  the  equation  of  a 
curve. 

"^    We  present  the  equation  of  the  circle  in  another  form  also, 
viz.  when  its  center  does  not  lie  at  the  origin  of  the  coordi- 


18 


CALCULUS 


[Ch.  I. 


nates  (Fig.  9).     If  the  coordinates  of  the  center,  ON  and 
ilffiV,  be  equal  to  a  and  h  resx^ectively,  let  the  coordinates  of 

the  point  P,  arbitrarily  chosen,  be 

OQ  =  x 

and  PQ  =  y; 

if  MR  is  equal  and  parallel  ta  NQ^ 
we  obtain  directly  from  the  right 
triangle  MPR 

But  since         MR  =  NQ  =  OQ  -  0N=  x  -  a, 


and 


PR  =  PQ-RQ  =  PQ-  31N=  y-h. 


we  have,  by  substitution, 

(5)  (x-a)2+(2/-6)2  =  /-% 

the  equation  of  the  circle. 

Exercise.  Discuss  similarly  various  positions  of  P  in  the  figure  above, 
and  also  various  positions  of  M  in  the  different  quadrants,  as  well  as 
various  magnitudes  of  the  radius,  including  cases  in  which  the  circle 
cuts  one  or  both  axes ;  making  a  figure  for  each  case,  and,  with  due 
regard  to  algebraic  signs,  always  reaching  equation  (5)  as  final  result. 

As  a  corollary  to  the  above,  we  can  deduce  the  formula 
which  gives  the  distance  between  two  points  whose  coordinates 
are  known ;  equation  (5)  yields  this  formula  at  once.  It 
states  that  the  distance  r  of  the  point  M  from  the  point 
P  is  represented  by  the  square  root  of  the  left-hand  member 
of  equation  (5).  If,  for  reasons  of  symmetry,  we  substitute 
a^  and  h-^  for  x  and  z/,  our  equation  gives  as  the  distance 
between  two  points  with  the  coordinates  (^  6)  and  (a^,  h^ 


(6) 


r^={a^-a)^+{h^-h)' 


I 


4.] 


THE  ELEMENTS   OF  ANALYTIC  GEOMETRY 


19 


Example.     The  square  of  the  distance  between  the  points  P^  and  P^ 
(Fig.  10)  whose  coordinates  *  are  (3,  4)  and  (2,  1)  amounts  to 

(3  -  2)2  +  (4  -  1)2,  or  10. 

If  we  conceive  the  line  P^P^  to  be 
moved  four  units  of  length  towards  the 
left,  so  that  it  assumes  the  position 
the  coordinates  of  the  points 
-  1,  4)  and  (-2,  1),  and 
the  square  of  the  distance  between 
them  proves  to  be,  as  required. 


i?j/i2   are    ( 


(-l-(-2))2+(4-l)2 
=  (2  -  1)2  +  (4  -  1)2  =  10. 


— 5-^— -5— 

---4-------p--- 

----i------------4---- 

=i=====l==== 

:::-i-^i::::::^: 

Fig.  10. 


Our  formula  holds,  therefore,  even  when  the  coordinates  of  the  points 
have  negative  values.  The  reason  for  this  lies,  of  course,  in  the  fact 
that  the  same  rules  of  calculation  apply  to  both  positive  and  negative 
quantities. 

EXERCISES  III 

1.  Find  the  equations  of  the  circles  having  the  following  centers  and 
radii  (the  point  given  and  the  number  following  it  being  respectively 
center  and  radius) : 

(v.)   (0,5),  4; 


(i.)  (-%4),6; 

(ii.)  (-2,  5),  2; 

(iii.)  (-2,4),3i 

(iv.)  (3,  -1),2; 


(vi.)   (0,  -  2),  I 
(vii.)   (4,0),  21; 
(viii.)   (0,  0),  1; 


(ix.)  (-2,0),  6; 

(x.)  (&,  c),  2; 

(xi.)  (m,  n),  k; 

(xii.)  (4ai,  -2  6),  3  A. 


2.    Find  the  distances  between  the  following  pairs  of  points: 
(i.)   (2,3),  (4,5);  (iv.)   (8,  2),  (- 4,  3); 

(ii.)   (-  2,  4),  (1,  0);  (V.)   (a,  -  a),  (h,2b); 

(iii.)    (1,  -  4),  (-  2,  -  3);  (vi.)    (a,  b),  (b,  a); 

(vii.)   (r,  sinct),  (r,  cos  a). 


*  The  unit  of  length  is  equal  to  two  of  the  spaces  into  which  the  ic-axis  is 
divided  in  the  fio;ure. 


20 


CALCULUS 


[Ch.  Ic 


3.    Show  that  the  coordinates  of  the  middle  point  of  the  straight  line 
joining  the  points  (a,  h)  and  (c,  d)  are 

a  +  g 
2 

and  ^-+A. 


(Construct  figures  variously,  with  given  points  lying  in  various  quad- 
rants.) 

4.  Find  coordinates  of  the  middle  points  of  the  lines  joining  each  pair 
of  points  in  2. 

5.  Find   the   equations   of   circles   each   passing  through  the  point 
(7,  —3),  and  having  as  centers  respectively  the  various  points  given  in  2. 


Art.  5.  The  equation  of  the  parabola.  Geometrically, 
the  parabola  is  the  locus  of  all  points  which  ay^e  equidistant 
from  a  fixed  point  and  a  fixed  straight 
line.  If  P  (Fig.  11)  be  any  point  of 
the  parabola,  F  the  fixed  point,  d  the 
fixed  straight  line,  and  PD  the  dis- 
tance of  the  point  P  from  (i,'the  con- 
dition that  defines  the  parabola  is 
expressed  by  the  equation 


(1) 


PF=PI), 


In  order  to  express  the  equation  of  the 
Fig.  11.  parabola  in  the  simplest  possible  form, 

we  choose,  as  the  a;-axis,  the  perpen- 
dicular FL  let  fall  from  F  on  6?,  and  the  middle  of  the  line 
FL^  as  the  origin  of  the  system  of  coordinates.  The  dis- 
tance FL  is  denoted  by  jt?,  and  is  called  the  parameter  of 
the  parabola^  If  x  and  y  are  the  coordinates  of  the  point  P, 
it  follows  that 


6.]  THE  ELEMENTS  OF  ANALYTIC  GEOMETRY  2l 


(2)  FF'^F^  +  P(^={x-&)+y' 

and,  by  comparison  with  (1), 


^ '  +  /  = 


-Ml 


or  p!;^-px-^=^-h  ^^  =  of-h  px  +  #, 

and,  finally,  '' 

(3)  2/2^2  ^^^ 

This  is  the  equation  of  the  parabola. 

We  add  that  the  point  F  is  termed  the  focus,  and  the 
straight  line  d  the  directrix  of  the  parabola. 

Exercise.  As  in  the  preceding  article,  discuss  various  positions  of 
P,  making  figure,  and  always  reaching  equation  (3)  as  final  result. 

The  question  arises,  how  can  the  form  of  the  .parabola  be 
deduced  from  its  equation  ?  It  is  perfectly  evident  that, 
if  X  has  a  negative  value,  y'^  must  also  be  negative.  But 
there  are  no  real  numbers  whose  square  is  negative,  and 
hence  the  points  of  the  parabola  can  lie  only  on  the  right 
side  of  the  «/-axis.     If 

a;  =  0,  then  y  =  0  -^ 

i.e.  the  parabola  passes  through  the  origin.  If  x  be  given 
any  positive  value,  as,  for  instance, 

X  =  0Q\ 

the  equation  furnishes  two  different  values  for  ?/,  but  differ- 
ing only  in  sign  ;  the  corresponding  points  are  P'  and  P'^ 
which  are  situated  at  the  same  distance  from  Q'  above  and 


2^  CALCULUS  [Ch.  1. 

below  the  axis  of  abscissae.  This  is  true  for  every  value  of 
X ;  the  points  of  the  locus  are  therefore  arranged  in  pairs 
symmetrically  with  reference  to  the  axis  of  abscissae  ;  accord- 
ingly, the  a;-axis  is  denominated  an  axis  of  symmetry  of  the 
parabola. 

We  remark  further  that,  if  greater  and  greater  values  be 
given  to  x^  the  values  of  ?/  increase  continually  ;  the  farther 
the  parabola  extends  from  the  a:-axis,  the  more  it  spreads  out. 
This  gives  us  a  preliminary  idea  of  the  form  of  the  parabola. 

On  inspection,  we  see  that  the  curve  constructed  in  Art.  3 
(p.  14)  is  a  parabola,  whose  parameter  is  equal  to  unity. 

Art.  6.  The  equation  of  the  straight  line  through  the 
origin.  We  first  deduce  the  equation  of  the  straight  line 
on  the  assumption  that  it  passes  through  the  origin  of  a 
system  of  rectangular  coordinates  ;  in  which  case  it  may 
be  defined  as  a  locus  such  that  the  angle  fo^^med  with  the 
X-axis  hy  the  (unterminatecT)  straight  line  joining  each  point 
of  the    locus    to    the    origin   remains  constant.      This    angle, 

however,  must  be  defined  more  ex- 
actly, since  two  straight  lines  can 
form  different  angles  wdth  each 
other.  Accordingly,  we  establish 
<  the  following  general  conditions : 
If  the  positive  portion  of  the  axis 
of  abscissae  be  rotated  counter-clock- 

FiG.  12. 

wise  about  0,  (Fig.  12)  as  a  center 
and  through  an  angle  of  90°,  it  comes  into  the  position  of 
the  positive  portion  of  the  axis  of  ordinates.  This  direction 
of  rotation  is  called  positive,  and  we  are  to  understand  by 
the  angle  which  a  straight  line  g  makes  with  the  a;-axis, 
that  angle  through  which  the  positive  a:-axis  must  be  rotated 


5-6.] 


THE  ELEMENTS  OF  ANALYTIC  GEOMETBY 


23 


in  the  positive  direction  around  the  point  of  intersection  of 
g  and  the  a:-axis  until  it  coincides  with  g ;  in  Fig.  12  this 
is  the  angle  (f).  The  line  g  is  capable  of  two  directions,  each 
measured  from  the  vertex  (f>  in  opposite  directions.  The 
positive  direction  is  measured  from  the  vertex  toward  that 
part  of  the  line  g  with  which  the  positive  part  of  the  a:-axis 
first  coincides  in  its  rotation  counter- 
clockwise. (In  the  figure  this  is 
shown  by  an  arrow.) 

If  g  (Fig.  13)  be  the  straight  line 
whose  equation  we  are  to  find,  -  we 
have,  to  start  with,  the  definition 
that  the  line  connecting  any  point  P 
with   0   always  makes  the   angle   a 

wdth  the  axis  of  abscissae.  If  we  designate  by  x  and  g  the 
coordinates  of  P,  it  follows  directly  from  the  right  triangle 
PO^that  V     * 


Fig.  13. 


or 


tan«  =  ^  = 


(1)  y  =  Qc  tan  a, 

which  is  the  required  equation  of  the  straight  line. 


EXAMPLES 

y  —  X  —  0, 

y  =  x, 
tan  a  =  1 ; 

«  =  45°. 
2/  +  a:  =  0, 

y  =  -^x 
tan  a  =  —  1 ; 
a  =  135°. 

These  two  lines  bisect  the  angles  which  the  axes  of  the  coordinates  form 
with  each  other. 


1.    The  equation 
or 

represents  a  line  for  which 
that  is, 
Likewise 
or 

is  a  straight  line  for  which 
that  is  to  say, 


24  CALCULUS  [Ch.  I 

2.  The  equation  y  =  0 

represents  a  straight  line  passing  through  0  and  having 

tan  a  =  0, 

and  hence  a  =  0; 

in  other  words,  the  line  is  the  axis  of  abscissae.  The  law  expressed  by 
the  equation 

y  =  0 

means  simply  that  the  line  is  the  locus  of  all  points  whose  ordinates  are 
equal  to  zero,  and  this  is  equivalent  to  saying  that  these  points  lie  in 
the  X-axis.     In  the  same  way 

x  =  0 

is  the  equation  of  the  y-axis ;  that  is  to  say,  the  equation  of  the  locus 
of  all  such  points  as  have  abscissae  equal  to  zero  (p.  10). 

3.  The  equation 

y  =y/3  '  X 

represents  a  straight  line  passing  through  0,  and  making  an  angle  of  60° 
with  the  a:-axis. 

Art.  7.   The  equation  of  any  straight  line.    If  the  straight 
line  (Fig.  14)  has  any  position  whatever  with  reference  to 

the  coordinate  axes,  and  if,  (i.) 

Gr  is  its  point  of  intersection  with 

the    ?/-axis,    (ii.)    a   is   the  angle 

which  it  makes  with   the  a;-axis, 

and  (ill.)  the  distance  OGr  is  equal 

to  J,  we  can  define  it  by  saying 

Fig,  14,  that  the   line   connecting   any  of 

its  points  P  with  Gr  has  the  same 

angle  of  inclination,  a,  with  the  a^-axis.      \i  PQ  and  GN  be 

perpendicular  and  parallel  respectively  to  the  a:-axis,  the  angle 

PGN  is  equal  to  «,  and  in  the  right-angled  triangle  PGN^ 

PN     y-h 

(1)  tan«  =  — =  L, 


G 

Y             . 

Nl 

^y^\ 

Y 

X            0 

( 

2      ^ 

■ 


6-7]  THE  ELEMENTS   OF  ANALYTIC  GEOMETRY  25 

where  x  and  y  are  the  coordinates  of  P ;  hence  its  follows  that 

(2)  y  =  ^  tan  a  +  5, 

which  is  the  required  equation  of  the  straight  line.  It'  is 
customary  to  denote  tan  a  by  w,  so  that  the  equation  assumes 
the  form, 

(3)  y  =  mno  +  &. 

The  equation  of  every  straight  line  *  is  of  this  form,  vari- 
ous equations  differing  from  one  another  only  in  the  values 
of  m  and  5,  which  depend  upon  the  positions  of  the  lines. 
For  example,  if  the  straight  line  passes  through  the  origin, 
its  equation  becomes  simplified  into 

(4)  y  =  mx\ 

quite  in  accordance  with  the  foregoing  results.  Conversely, 
a  straight  line  is  represented  by  every  equation  of  the  form 

y  —  mx  +  h^ 

no  matter  what  the  values  of  m  and  h  may  be.  For  there 
must  always  be  a  point  Cr  determined  by  5,  whatever  may 
be  the  number  that  h  stands  for;  and  likewise  for  every 
number  m  there  must  exist  an  angle  a,  so  that 

tan  a  =  m; 

if,  then,  we  form  the  equation  of  the  straight  line,  cutting 
off  a  distance  b  on  the  «/-axis,  and  making  an  angle  whose 
tangent  is  m  with  the  a;-axis,  we  shall  have 

y  =  mx  -f-  b. 


*  Provided  the  line  is  not  perpendicular  to  the  x-axis.  We  have  already 
seen  (Ex.  5,  iii.,  p.  11)  that  the  equations  of  all  straight  lines  perpendicular 
to  the  jc-axis  are  of  the  form  x  =  c. 


26  CALCULUS  [Ch.  I. 

But  this  is  precisely  the  equation  under  consideration,  and 
we  see,  therefore,  that  it  represents  a  straight  line  irrespec- 
tive of  the  values  of  m  and  h. 

We  do  not  think  it  superfluous  to  furnish  direct  proof  that  our  con- 
clusions and  results  are  not  changed  when  b  or  m  have  negative  values ; 
that  is,  when  the  position  of  the  straight  line 
with  reference  to  the  coordinate  axes  leads  to 
a  figure  apparently  different.  For  the  straight 
_y^  line  drawn  in  Fig.  15,  it  follows  from  the  tri- 
angle PGi\^  that 


N 


I^  tanl80-«  =  ^=^5+^. 

NG  NG 


Fig.  15.  But  in  this  case  x  and  h  are  negative  quanti- 

ties and  accordingly,  (p.  9),  the  length  ot'QN 
is  expressed  by  —  I,  and  that  of  NG  by  —  x*     Furthermore, 
tan  180  —  a  =  —  tan  «, 

and  hence  —  tan  a  = > 

—  X 

or  y  =  X  tan  a  +  h  =  mx  +  b. 

The  reason  for  the  general  validity  of  the  results  lies  in  the  circum- 
stance that  the  rules  of  calculation  with  positive  and  with  negative  quan- 
tities, as  well  as  the  trigonometric  formulae  for  acute  and  for  obtuse 
angles,  are  the  same ;  so  that,  even  though  the  figures  may  be  different 
for  diiferent  positions  of  the  straight  lines,  their  properties,  their  laws  — 
and  everything  hinges  upon  these  alone — remain  the  same  in  all  cases. 
In  what  follows  we  may  accordingly  omit  calling  special  attention  in 
each  case  to  the  generality  of  our  formulae. 

Exercise.     Deduce  the  equation 

y  =  mx  +  b 
from  other  possible  figures,  noting  also  that  the  figure  is  altered  if,  while 
the  line  remains  unmoved,  the  variable  point  P  be  taken  in  a  different 
quadrant. 

*  Regarded  as  coordinates,  QN  and  NG  represent  negative  numbers,  viz. 
h  and  X.  Regarded  as  purely  geometric  magnitudes,  their  lengths  are  positive, 
viz.  —  b  and  —  x.  It  often  happens,  as  above,  that  we  first  discuss  problems 
geometrically,  all  the  lines  involved  being  regarded  as  positive  magnitudes, 
irrespective  of  position,  and  then  express  these  positive  magnitudes  in  terms 
of  the  coordinates  (positive  or  negative)  which  the  lines  represent. 


I 


7-8.]  THE  ELEMENTS    OF  ANALYTIC  GEOMETRY  27 

Art.  8.  Every  equation  of  the  first  degree  represented  by 
a  straight  line.  We  deduce  from  the  foregoing  the  impor- 
tant conclusion  that  every  equation  of  the  form 

(1)  Ax  +  By+C=0, 

wherein  A^  B,  and  0  are  any  positive  or  negative  numbers 
whatever,  is  the  equation  of  a  straight  line.  For,  dividing 
the  equation  by  ^  *  and  transposing,  we  get, 

A         O 
y  =  'B''-W 

and  this  is  the  equation  for  a  straight  line  in  which 

(2)  tan«  =  -4' 

and(3)''  ^  =  -^' 

B 

We  now  see  that  the  equation  of  p.  12, 

X  +  y  =  4,  or  y  =  —  X  -\-  4:^ 

represents  a  straight  line,  which  cuts  off  on  the  axis  of  ordi- 
nates  a  distance  equal  to  4  units,  and  forms  with  the  axis 
of  abscissae  an  angle  such  that  tan  «  =  —  1,  i.e.  an  angle  of 
135°,  just  as  is  shown  in  Fig.  6. 

Equation  (1),  being  general  in  character  and  containing 
only  the  first  powers  of  x  and  y,  is  called  the  general  equa- 
tion of  the  first  degree ;  from  (3,  p.  25)  and  (1)  together  we 
see  that  the  straight  line  is  the  ciraphic  equivalent  of  tJw 
general  equation  oj  the  ^^f^,,  d^ffree.  The  equation  of  the 
tirst  degree  is  accordingly  often  called  the  linear  equation. 


*  This  implies  that  B  is  not  zero.     When  B  is  zero,  the  equation  takes 

the  form  x  =  —  — ,  or  x  =  c,  which  has  been  discussed  previously  (pp.  11,  25). 
A 


28  CALCULUS  [Ch.  I. 

EXAMPLES 

1.  In  the  equation 

and  tan  «  =  2 ; 

(a  being  accordingly  equal  to  about  63°  26'). 

2.  In  the  equation 

b  =  2, 
and  tan  «  =  ^, 

(or  a  =  about  18°  26'). 

The  two  lines  can  be  drawn  by  the  aid  of  these  data. 

Art.  9.  The  intercepts.  Use  is  often  made  of  another 
method  to  find  out  the  position  of  a  straight  line  from  its 
equation.  Tlie  angle  a  is  not  employed  in  it,  since  angles 
are  inconvenient  in  actual  constructions ;  but  any  two 
points  on  the  line  are  sought ;  and  these  determine  the 
position  of  the  straight  line.  The  points  which  can  be 
found  most  conveniently  are  those  at  which  the  lines  inter- 
sect tlie  axes.  If  the  equation  of  the  straight  line  is  given 
in  the  general  form 

(1)  Ax-\-B^  +  O=0, 

these  points  are  obtained  in  the  following  manner  :  The  point 
of  intersection  with  the  axis  of  ordinates  is  the  point  whose 
abscissa  is  equal  to  zero  ;  we  find  it  by  making  x^  in  the 
above  equation,  equal  to  zero.     This  gives  the  point 

(2)  x  =  0,   2/  =  -|- 

Similarly,  the  point  of  intersection  with  the  axis  of  ab- 
scissse  is  the  point  whose  ordinate 


8-10.]        THE  ELEMENTS   OF  .ANALYTIC  GEOMETRY  29 

this  condition  yields  the  equations 

(3)  2/=0,   x  =  -^ 

The  distances  from  the  origin  to  the  points  of  intersection 
with  the  axes  are  called  the  intercepts  on  the  axes. 

Example.     The  points  of  intersection  of  the  straight  Kne 
5^-72/ +  2  =  0, 
with  the  axes  are  (0^  f )  and  (  —  |,  0),  and  the  intercepts  are  —  f  and  f . 

EXERCISES  IV 

1.  Find  the  intercepts  on  the  axes,  and  the  tangent  of  the  angle  made 
with  the  a:-axis  by  the  straight  lines  which  have  the  following  equations : 

(i.)  3a: -2^-4- 7  =0;    (iv.)  3a;  =  9?/ -  2;  (vii.)  2?/ +  3a;  =  0  ; 

(ii.)  y  =  2a;  -  5;  (Y.)2x  =  ^y\  (viii.)  ?/ +  3a;  -  5  =  0; 

(iii.)  2?/  + 5a;-4  =  0;    (vi.)  5a;  +  9y  +  4  =  0;     (ix.)  ^z/ -  a;  +  2  =  0. 

2.  Examining  Fig.  14,  we  see  that  the  straight  line  starts  from  the 
first  quadrant,  passes  through  the  second  into  the  third ;  in  Fig.  15  the 
line  passes  from  the  second,  through  the  third,  into  the  fourth  quadrant. 
Through  what  quadrant  does  each  line  of  exercise  1  above  pass  ?  (An- 
swer by  inspection  of  the  equations.) 

3.  What  are  the  equations  of  the  straight  lines  parallel  respectively  to 
those  of  1,  and 

i.  Passing  through  the  origin ; 

ii.  Having  the  intercept  5  on  the  ?/-axis? 

Art.  10.  Gay  Lussac's  Law.  According  to  the  law  of 
Gay  Lussac,  gases  possess  the  following  property :  If  their 
volume  be  kept  constant,  the  pressure  necessary  to  confine 
them  to  that  constant  volume  increases  proportionally 
to  the  temperature,  and,  indeed,  if  p^  be  the  pressure  at 
0°  Centigrade,  the  increase  for  one  degree  is  r^,  and 
hence  at  t°  the  pressure  must  be 


i'=i'o+fe)^=Po(n-2Y3> 


30  CALCULUS  [Ch.  I. 

If,  to  simplify  matters,  we  assume  the  value  of  the  pressure 
Pq  to  be  1,  this  formula  becomes, 

This  equation  is  an  equation  of  the  first  degree  in  t  and  p ; 
if  we  substitute  for  p  and  t^  y  and  x^  respectively,  so  that 
the  above  equation  becomes, 

^  =  ^  +  273' 

it  may  be  represented  by  a  straight  line ;  this  straight  line 

(Fig.  16)  is  the  graphic  represen- 
Y  tation  of  the  law,  and  shows  with 

great  clearness  that  the  pressure 
varies  continuously  with  the  tem- 
perature.* 

Art.  11.  Problems  on  the  straight  line,  I.  Wliat  is  the 
position  of  a  straight  line  whose  equation  is 

(1)  -  +  f  =  l? 

^  ah 

We  determine  as  before  the  points  at  which  it  intersects 

the  axes.     In  order  to  obtain  the  intercept  on  the  ^/-axis, 

we  put 

x  =  0 

in  the  above  equation,  and  find 

y  =  h', 

*  In  order  to  obtain  the  correct  position  of  the  straight  line,  OA  has  to 
be  taken  273  times  as  great  as  OB.  This  is  not,  however,  convenient  for 
the  sketch,  and  the  above  figure  is  but  an  approximate  representation  of 
the  line.  A  similar  method  must  always  be  employed  when  the  numerical 
values  of  the  coordinates  are  in  too  unfavorable  relations  for  an  accurate 
drawing. 


10-11.]      TEE  ELEMENTS   OF  ANALYTIC  GEOMETRY  31 

let  this  point  be  B  (Fig.  17).     Similarly,  A^  the  point  of 
intersection  with  the  a;-axis  is  found  to  have  the  coordinates, 

X  =  a, 

The  quantities  a  and  h  are,  accordingly, 
the  intercepts  on  the  axes. 

The  form  -  +  l  =  l 

is  known  as  the  symmetric  equation  of  the  straight  line. 

EXAMPLES 

1.  By  writing  the  equation 

X  +  y  ^  4 
(p.  12)  in  the  form 

it  appears  that  the  straight  line  which  it  represents  cuts  off  a  distance  4 
on  each  axis. 

2.  In  the  same  way  we  find  for  the  equation 

4:x  +  Sy-2  =  0, 
when  transformed  into 

the  intercepts  a  =  ^  =  ^f 

and  b  =^. 

II.  To  determine  the  equation  of  a  straight  line,  having  a 
given  direction  with  reference  to  the  axis  of  ahscissoe,  and 
passing  through  a  given  point  P,  whose  coordinates  are  x^ 
and  y-^. 

According  to  p.  25,  the  equation  of  every  straight  line 
has  the  form 
(2)  y=^mx  +  h) 


32  CALCULUS  [Ch.  I. 

to  find  the  equation  of  a  given  straight  line,  it  is  necessary 

to  find  the  values  of  m  and  b  which  correspond  to  that  line. 

Let,  then,  equation  (2)  be  the  equation  of  our  straight  line ; 

then  m  is  known,  viz. 

m  =  tan  a, 

a  being  the  given  angle,  while  b  has  a  definite  but  as  yet 
unknown  value.  Since  the  coordinates  of  the  point  P  must 
satisfy  the  equation  of  the  straight  line,  it  follows  that 

(3)  7/^  =  mx^-{-b; 

b  alone  is  unknown  in  this  equation,  and  can  therefore  be 
found  from  it  and  substituted  in  the  first  equation.  By  so 
doing  we  have,  in  principle,  solved  our  problem.  But  we  can 
obtain  the  required  equation  in  a  somewhat  different  way. 
To  calculate  b  from  equation  (3),  and  to  substitute  its  value 
in  equation  (2),  is  to- eliminate  b  from  equation  (2)  by  means 
of  equation  (3).  The  simplest  way  to  effect  this  elimina- 
tion is  to  subtract  one  equation  from  the  other;  we  then  find 

(4)  ^-7j^  =  m(x-x^}, 

and  this  is  the  required  equation  of  the  straight  line. 

III.    To  find  the  equation  of  a  straight  line  which  passes 
through  tivo  given  points  whose  coordinates  are  x^y^  and  x^j^. 

The  required  equation  of  the  straight  line  must  have  the 
form 

(5)  y  =  mx  4-  b^ 

where  m  and  b  are  definite,  although  as  yet  unknown,  quan- 
tities whose  values  are  to  be  calculated.  Since  the  coordi- 
nates x^^  y^  and  x^^  y.^  satisfy  the  equation  of  the  straight 
line,  we  obtain 


11.]  THE  ELEMENTS   OF  ANALYTIC  GEOMETRY  33 

(6)  y^  =  mx^-\-b; 

(7)  ^2  =  ^^2  +  ^' 

and  these  are  tlie  two  equations  from  which  we  are  to  cal- 
culate the  values  of  m  and  5,  and  then  substitute  them  in 
equation  (5).  This  means,  in  other  words,  that  we  must 
eliminate  m  and  b  from  our  three  equations,  (5),  (6),  and 
(7).  The  simplest  way  is  the  following:  We  subtract  the 
third  equation  from  the  first  and  from  the  second  equation, 
and  thus  obtain 

2/  ~  ^2  =  ^(^  -  ^2)' 

y\-  yi  =  ^<^i  -  ^2)  ' 

and  by  dividing  the  first  of  these  equations  by  the  second, 
we  get  finally 

(8)  y~yi,  =  x  —  Xj^ 

y\ ~yi    ^i~ ^2 

as  the  required  equation.* 

EXAMPLES 

1.  The  equation  of  the  straight  line  that  passes  through  the  points 
(2,  1)  and  (-3,  4)  reads: 

y  -  4  _  a;  +  3 
i  -  4  ~  2  +  3' 

or,  in  a  simplified  form,       3  x  -{■  5  1/  —  11  =  0. 

2.  The  equation  of  the  straight  line  passing  through  the  points  (3,  2) 
and  (—  3,  —  2)  is 

22:-3?/  =  0; 

the  line  accordingly  passes  through  the  origin. 

*  Of  course  the  equation  can  also  be  put  into  a  form  from  which  the 
values  of  m  and  b  can  be  directly  read  off.  A  simple  transformation  of 
equation  (8)  gives: 

y  _  yi  -  y2  ^  _^  mh  -  xm 

X\  —  X2.  Xi—X^* 


34 


CALCULUS 


[Ch.  I. 


IV.  To  find  the  equation  of  a  straight  line,  given  the  length, 
p,  of  the  perpendicular  on  it  fr-om  the  origin,  and  the  angle  a 
which  that  perpe:ndicular  makes  with  the  x-axis. 

y 


Fig.  19. 


In  both  figures  (18,  19),  OA  =  p;  BOA  =  a.* 

OA  =  DA-  OB 


Then       OA  =  OB -\-  BA 
=  OB  +  PC. 

But      PBC=a, 

and  hence, 

PC  =  PB  sin  OL  =  y  sin  a. 

Likewise, 

OB  =  OB  cos  a  =  X  cos  a. 


=  PO-OB. 

But 

PBC=BOB  =  a-im'', 

and  hence, 

PC=PB^inPBC 

=  PJ5sin(a- 180°) 
=  -  y  sin  (a  -  180°) 
=  y  sin  a. 

Likewise, 

OB  =  OBgo^BOB 

=  X  cos  (a  -  180°) 
=  —  X  cos  ct. 
(9)  .•.  ^  =  2/  sin  a  +  a?  cos  a. 

This  is  known  as  the  normal  equation  of  the  straight  line. 

*  As  OA  is  a  terminated  portion  of  a  straight  line,  we  regard  as  the  angle 
between  OA  and  the  x-axis,  the  angle  through  which  the  positive  portion  of 
the  5c-axis  revolves  in  the  positive  sense  until  it  comes  for  the  first  time  into 
coincidence  with  OA. 


11-12.]      THE  ELEMENTS  OF  ANALYTIC  GEOMETRY  35 

Art.  12.   Concerning  the  nature  of  a  general  equation. 

In  the  various  forms  of  the  equation  of  the  straight  line,  which  we 
have  considered,  there  enter  constant  coefficients.  When  these  constants 
are  given  different  values,  different  straight  lines  are  represented,  and 
every  possible  straight  line  may  be  represented  by  giving  the  constant 
coefficients  proper  values.  We  may  start  from  a  given  straight  line  ful- 
filling given  conditions  (such  as  to  pass  through  two  given  points),  and 
set  ourselves  the  problem  to  find  the  equation  which  will  represent  this 
line,  i.e.  to  find  the  particular  values  which  must  be  given  to  the  con- 
stants of  the  general  equation  (say  y  =  mx  +  h)  in  order  that  it  may  rep- 
resent the  line  under  consideration.  We  have  solved  several  problems 
of  this  kind.  In  doing  so  we  regarded  the  constants  as  unknown  quan- 
tities to  be  determined.  This  notion  of  constant  quantities  which  may 
vary  (and  which  may  indeed  assume  all  possible  values)  is  sometimes 
perplexing  to  the  student.  The  fact  is  that  the  coefficients  are  constant 
for  every  specific  line,  but  vary  from  line  to  line,  while  the  variables  x 
and  y  assume  for  each  line  all  the  values  compatible  with  the  relation 
established  between  them  by  the  equation.  The  notion  of  arbitrary  con- 
stants in  a  general  equation  is  of  such  fundamental  importance  that  we 
add  an  illustration  which  may  make  it  clearer. 

A  printed  form  of  mortgage  can  be  bought  at  the  law  stationer's. 
It  contains  blanks  for  the  names  of  the  mortgagor  and  mortgagee,  for 
the  exact  description  of  the  piece  of  land  mortgaged,  for  the  considera- 
tion and  the  amount  of  the  mortgage,  for  the  date  when  it  is  executed, 
and  the  time  it  has  to  run.  Until  these  blanks  are  properly  filled  out 
it  is  no  particular  mortgage,  but  it  may  become  any  mortgage  by  filling 
out  the  blanks  suitably.  Yet  all  the  properties  of  a  mortgage  as  a 
mortgage  can  be  learned  from  this  printed  form ;  its  general  legal  aspects 
and  force  can  be  determined  as  well,  perhaps  better,  than  if  it  were 
specialized  into  a  particular  mortgage.  The  blank  form  is  the  general 
mortgage  —  all  the  characteristics  common  to  all  mortgages  can  be  learned 
from  it,  —  and  when  this  general  mortgage  is  once  thoroughly  understood 
it  is  a  trifling  matter  to  draw  up  a  specific  mortgage.  We  may  also 
discuss,  if  we  please,  the  various  kinds  of  mortgages  that  may  arise  from 
various  styles  of  filling  out  the  blanks,  such  as  farm  mortgages,  mort- 
gages on  vacant  city  lots,  on  improved  city  real  estate  and  the  like. 

The  case  of  a  general  equation,  for  instance  y  —  mx  +  6f  is  quite  analo- 
gous. This  contains  two  blanks,  one  denoted  by  m,  for  the  tangent  of  a, 
the  angle  which  the  straight  line  represented  by  the  equation  makes  with 
the  X-axis ;  the  other  denoted  by  &,  for  the  length  of  the  intercept  on  the 


86  CALCULUS  [Ch.  I. 

2/-axis.  When  the  blanks  m  and  h  are  not  filled  out,  i.e.  have  no  specific 
numerical  value,  this  equation  represents  no  particular  straight  line,  but 
it  may  represent  any  straight  line  by  filling  out  the  blanks  suitably.  It 
is  the  general  equation  of  the  straight  line ;  from  it  in  the  blank  form 

y  =  mx  +  h 

all  the  properties  common  to  all  straight  lines  can  be  learned,  and  when 
this  general  form  is  thoroughly  understood,  the  equation  of  specific  straight 
lines  can  be  written  out  at  will.  We  may  also  discuss  the  classes  of 
straight  lines  we  obtain  by  filling  out  blanks  in  various  styles;  thus,  if  we 
fill  the  blank  h  with  zero,  the  line  will  pass  through  the  origin,  no  mat- 
ter how  the  blank  m  is  filled  out.     The  equation 

y  =  mx 

is  then  the  blank  form  for  the  equations  of  all  straight  lines  passing 
through  the  origin,  or  it  is  the  general  equation  of  all  such  lines. 
Similarly 

y  =  x  +  h 

is  the  general  equation  of  all  lines  making  an  single  of  45"  with  the 
axis  of  X. 

We  have  spoken  above  oi         y  —  mx  -\-h 

as  the  general  equation  of  the  straight  line.  This  is  not  meant  to  imply 
that  there  may  not  be  other  general  equations  of  the  straight  line.  Just 
as  there  may  be  more  forms  than  one  for  a  mortgage,  such  that  all 
possible  mortgages  may  be  drawn  up  according  to  either  one,  so  there 
may  be,  and  in  fact  are,  more  forms  than  one,  of  equations  such  that  all 
possible  straight  lines  may  be  represented  by  any  one  of  them,  by  filling 
up  the  blank  coefficients  properly. 

EXERCISES  V 

1.  Plot  the  straight  lines  represented  by  the  following  equations : 

(ii.)  y  =  2x-2;  4 

(vi.)  X  cos  40°  +  y  sin  40°  =  3 ; 

4  ~  2  ^  "^ '  (vii.)  X  cos  112°  +  y  sin  112°  =  2 ; 

(iv.)  2  X  -  3  y  =  1 ;  (viii.)  x  cos  243°  +  y  sin  243°  =  5. 

2.  Find  the  general  equation  of  all  straight  lines  through  the  point 
(2,  4),  likewise  find  the  general  equation  of  all  straight  lines  through  the 
point  (—1,  5),  and  also  of  those  through  the  point  (  —  2,  —  2). 


(111-)  T-o  =  i; 


12.]     THE  ELEMENTS   OF^m^^fflC  GEOMETRY  37 


3.  Find  the  equation  of  the  particular  straight  line  through  each  of 
the  points  of  2  above,  which  makes  an  angle  of  120°  with  the  a;-axis ; 
likewise  of  the  straight  line  through  each  point  which  makes  an  .angle 
with  the  a:-axis  whose  tangent  is  —  2 ;  likewise  of  those  which  make  an 
angle  with  the  y-axis  whose  tangent  is  \. 

4.  What  is  the  general  equation  of  all  straight  lines  : 
(i.)  Parallel  to  the  ar-axis? 

(ii.)  Parallel  to  the  ^/-axis? 
(iii.)  Through  the  origin  V 
(iv.)  Making  an  angle  of  45°  with  the  x-axis? 
(v.)  Making  an  angle  of  —  45°  with  the  x-axis? 
(vi.)  Parallel  to 

(a)  y  =  4:X-2,  (c)  x  =  y, 

(b)  2x  +  3y  =  Q,  (d)  y  =  -^xl 
(vii.)   At  the  distance  3  from  the  origin  ? 

5.  Write  the  equation  of  the  straight  lines  passing  through  the  point 
(2,  1)  and  parallel  respectively  to  the  first  five  lines  in  1  above. 

6.  Find  the  equations  of  the  straight  lines  passing  through  thje  fol- 
lowing pairs  of  points : 

(i.)   (1,2),  (0,1);  (iv.)  (-6,4),   (3,5); 

(ii.)  (-  4,  3),  (-  2,  -  1)  ;  (V.)   (a,  a),   (b,  b)  ; 

(iii.)  (1,  -  1),  (  -  1,  1)  ;  (vi.)  (a,  6),  (b,  a)  ; 

(vii.)  (a,  b),  (c,  d). 

7.  Three  of  the  vertices  of  a  parallelogram  are  (—  4,  1),  (—  1,  —  6), 
(2,  3).     Find  the  equations  of  its  four  sides. 

8.  Deduce  the  normal  equation  of  the  straight  line  from  a  figure  in 
which  the  straight  line  passes 

(i.)  through  the  second  quadrant ; 
(ii.)  through  the  fourth  quadrant ; 
(iii.)   through  the  origin. 

9.  Denoting  by  d  the  distance  of  a  straight  line  from  the  origin, 
and  by  A  the  angle  which  the  perpendicular  from  the  origin  on  the 
straight  line  makes  with  the  x-axis,  write  the  equations  of  the  lines 
having: 

rf  =  5  2  4  3  2 

A   =  120°    45°    225°    270°    330° 


SB  CALCULUS  [Ch.  1. 

10.  Starting  from  the  general  equation  in  the  normal  form,  find  the 
equations 

(i.)   of  the  bisectors  of  the  quadrants ; 

(ii.)    of  all  lines  parallel  to  the  a;-axis ; 
(iii.)   of  all  lines  parallel  to  the  ^-axis. 

11.  A  triangle  has  as  its  vertices  the  points  (2,  3),  (1,  7),  (—4,  2). 
Find  the  equations  of  its  sides. 

12.  A  right  triangle  has  the  vertices  of  its  acute  angles  in  the  points 
(4,  6),  (  —  2,  —5),  and  the  other  sides  parallel  to  the  axes.  Find  the 
equations  of  its  sides,  and  its  area. 

13.  Find  the  equation  of  the  circle  which  has  the  points  (3,  5),  (—4, 
—  3)  as  extremities  of  a  diameter.  Use  this  result  to  name  the  third 
vertex  of  four  right  triangles  having  the  points  (3,  5)  and  (—4,  —  3)  as 
extremities  of  their  hypotenuse,  and  the  third  vertex  lying  in  turn  in 
each  of*  the  four  quadrants.  How  many  solutions  are  possible  ?  Make 
a  second  choice  of  the  four  third  vertices,  so  that  they  shall  be  the  four 
corners  of  a  rectangle.  Find  the  equations  of  the  diagonals  of  your 
rectangle. 

Art.  13.  Two  straight  lines.  If  two  straight  lines  are 
given,  we  are,  above  all,  interested  in  knowing  their  point 
of  intersection  and  the  angle  which  they  make  with  each 
other. 

Let  the  equations  of  the  two  straight  lines  be 

(1)  9/  =  mx  +  b  and  y  =  m'x  +  h'. 

Each  equation  is  satisfied  by  a  boundless  number  of  pairs 
of  values  of  x  and  ^,  and  indeed,  each  one  is  satisfied  by  the 
coordinates  of  any  of  its  points.  These  pairs  of  values  are 
generally  different,  but  there  is  necessarily  one  and  only  one 
pair  among  them  that  satisfies  both  equations,  and  that  is 
the  one  which  corresponds  to  the  point  of  intersection  of  the 
lines.  In  order  to  find  this  pair  of  values,  we  have  to  deter- 
mine by  the  ordinary  methods  the  values  of  the  unknowd 
quantities  x  and  i/  which  satisfy  both  equations  simultane- 


12-13.]      THE  ELEMENTS  OP  ANALYTIC  GBOMETBT 


39 


ously.  The  solving  of  two  equations  of  the  first  degree 
with  two  unknown  quantities  means,  then,  geometrically 
speaking,  the  finding  of  the  point  of  intersection  of  the  two 
lines,  which  are  represented  by  the  two  equations. 

Example.   The  three  straight  lines 

x  +  1  y  ^-11  =  0, 
ar  -  3  ?/  +  1  =  0, 

3x  +  ?/-7  =  0 

define  a  triangle  whose  vertices  are  the  points  of  intersection  of  these 
lines  taken  in  pairs,  viz. :  (2,  1),  (3,  —2),  (—4,  —1). 

The  angle  8  which  two  straight  lines  form  with  each  other 
is  to  be  understood  as  being  the  angle  which  their  positive 
directions  form.     If  a  and  a'  de- 
note the  angles  which   the   two 
lines  make  with  the  axis  of  ab- 
scissae, the  value  of  3  is 

8  =  ot'  —  a. 

Accordingly, 

tan  S  =  tan  (a'  —  a) 

(2) 


tan  a'  —  tan  a 


^-^ 


1  +  tan  a'  tan  a 


Fig.  20. 


If  we  now  substitute  m  for  tan  a,  and  m'  for  tan  a\  we 
have 


(3) 


tan  8  = 


m' 


m 


1  +  mm' 

Should  the  straight  lines  be  parallel,  then  8  as  well  as 
tan  8  must  be  equal  to  zero ;   that  is  to  say, 

(4)  m  =  m'. 


40  CALCULUS  [Ch.  I. 

If  the  two  straight  lines  are  perpendicular  to  each  other, 
the  magnitude  of  B  is  90°,  while  tan  8  becomes  infinitely 
large  ;  the  denominator  of  the  above  fraction  must  accord- 
ingly be  equal  to  zero, 

(5)  1  +  mm'  =  0  ; 

1 

or  m'  = •> 

m 

and  only  when  this  condition  is  fulfilled  can  the  two  lines 
be  perpendicular  to  each  other. 

Example.   The  angle  8  of  the  two  straight  lines 
3x  + ?/  -7  =  0 
and  x-3y  +  l=0 

is  found  to  be  90°  (the  triangle  mentioned  above  is  therefore  right- 
angled  at  the  point  (2,  1)  of  intersection  of  these  two  lines). 

Bemark.  We  have  just  seen  that  two  equations  of  the  first  degree  in  a; 
and  y  determine  the  position  of  the  point  at  the  intersection  of  the  straight 
lines  in  question.  Since  when  a  and  b  are  the  coordinates  of  a  point  P,  this 
can  be  expressed  by  the  equations 

(6)  X  =  a,  y  =  b, 

the  question  immediately  arises  as  to  whether  these  two  equations  can  be 
interpreted  in  the  way  indicated  above.     As  a  matter  of  fact  this  is  the  case. 

The  equation  y  =  b  i^  according  to  p.  25,  the  equation  of  a  straight  line 
for  which  m  =  0,  and  hence  tan  a  (as  well  as  a  itself)  is  equal  to  zero  ;  that  is, 
the  line  is  parallel  to  the  axis  of  abscissae  ;  further  (Fig.  14,  p.  24),  it  passes 
through  the  point  G  of  the  axis  of  ordinates,  for  which  OG  =  b;  it  is  accord- 
ingly parallel  to  the  ic-axis  and  is  at  a  distance  b  from  it.  The  equation 
y  =  6  is  the  locus  of  all  points  whose  ordinates  are  equal  to  &,  and  all  of  them 
lie  upon  this  parallel  to  the  x-axis.  In  a  similar  manner,  it  follows  that  x  =  a 
represents  a  straight  line  which  runs  parallel  to  the  y-axis  at  a  distance  a 
from  it ;  and  these  two  lines  intersect  in  that  point  whose  coordinates  are 
defined  by  the  equations  (6). 

Geometrically,  then,  the  use  of  coordinates  is  tantamount  to  -regarding 
every  point  of  a  plane  as  the  point  of  intersection  of  two  straight  lines  which 
are  parallel  to  two  fixed  axes.  If  all  possible  parallels  to  the  a^-axis  and  all 
those  to  the  y-axis  be  thought  of,  one  of  each  of  these  sets  of  parallels  passes 
through  any  given  point,  and  the  distances  of  these  lines  from  the  axes  are 
the  coordinates  of  the  points  in  which  they  intersect. 


13.]  THE  ELEMENTS   OF  ANALYTIC   GEOMETRY  41 

EXERCISES  VI 

1.  Find  the  intersections  of  the  following  pairs  of  lines : 

(i.)  2x-3?/  +  5  =  0,  ?/-4:r +  7  =  0; 

(ii.)  ^  +  1  =  1,  ^y  =  x', 

(iii.)  4a:-9?/  =  2,  3a:  +  2  7/  =  -5; 
(iv.)  ax  +  %  =  1,  hx  +  ay  =  1. 

2.  (i.)  What  is  the  general  equation  of  all  straight  lines  through  the 
point  (-1,7)  ? 

(ii.)    Through  the  point  (a,  —  3  «)  ? 

(iii.)  What  is  the  general  equation  of  all  straight  lines  parallel  to 
?/  =  4a:-3? 

3.  Show  that  the  general  equations  of  all  straight  lines,  perpendicular  to 

y  =  mx  +  h 

•  '^  x 

is  y  = he. 

711 

4.  (i.)  Write  the  general  equation  of  all  straight  lines  perpendicular 
to  each  of  the  lines  of  1  above. 

(ii.)  Write  the  equation  of  the  perpendicular  from  the  origin  to  each 
of  the  lines  of  1. 

(iii.)  Find  the  tangent  of  the  angle  between  each  of  the  pairs  of  lines 
inl. 

5.  Find  the  vertices  of  all  triangles  formed  by  the  four  lines : 

X  =-y, 

?^  +  10x4-  18  =  0, 
a:  =  —  16  —  4?/. 

6.  In  each  triangle  of  5  find  : 

(i.)  The  equation  of  the  perpendicular  from  one  vertex  (any  one)  on 
the  opposite  side. 

(ii.)  The  coordinates  of  the  foot  of  the  perpendicular, 
(iii.)  The  length  of  the  perpendicular. 

(iv.)  The  length  of  the  side  to  which  the  perpendicular  is  drawn, 
(v.)  The  area  of  the  triangle. 

7.  In  any  one  of  the  triangles  of  5  verify  by  forming  the  equations 
of  the  perpendicular  bisectors  of  the  sides,  the  geometrical  theorem  that 
the  perpendicular  bisectors  of  the  sides  of  a  triangle  meet  in  a  point. 


42 


CALCULtlS 


[Ch.  I. 


8.  By  considering  the  general  triangle,  whose  sides  are : 

aa;  +  %  +  c  ==  0^ 
dx  -\-  ey  +  f  =  0, 
gx  -\-  hy  -\-  k  =  0. 
prove  generally  the  theorem  verified  in  7. 

9.  Which  of  the  triangles  that  can  be  formed  by  combining  any  three 
of  the  following  lines  is  right  angled  ? 

x  =  2y  +  l, 
■      2y-]-x  +  5=0, 
Sy  +  (jx  =  15, 
y  =  2x  +  l. 

10.  What  are  the  equations  representing  the  sides  of  the  general 
right  triangle  ? 

11.  Verify  in  one  of  the  right  triangles  found  in  9  that  the  line  join- 
ing the  middle  of  the  hypotenuse  to  the  opposite  vertex  divides  the 
triangle  into  two  isosceles  triangles. 

12.  By  considering  the  general  right  triangle  as  found  in  10  prove 
generally  the  theorem  verified  in  11. 

Art.  14.  The  equation  of  the  ellipse.  We  define  the 
ellipse  as  the  locus  of  a  point  which  moves  so  that  the  sum  of 

its  distances  from  two  fixed 
points  has  a  constant  value. 
This  constant  value  we  indi- 
cate by   2  a. 

We  take  (Fig.  21)  the  line 
connecting     the      two     fixed 
points  J\  and  F^  as  the  axis 
of  abscissae,  and  the  perpen- 
dicular erected  at  the  middle 
of  F^F^  as  the   axis   of  ordi- 
nates.     The  distance  F^F^  is  designated  by  2  c,  and  r^  and 
7*2  represent  the  distances  of  any  point  in  the  ellipse,  as  F 
from  F^  and  F^y  so  that 


r^^M  oU'.. 


13-14.]      THE  ELEMENTS   OF  ANALYTIC  GEOMETRY  43 

(1)  r^  +  r^  =  2a; 
furthermore,  it  is  apparent  that 

(2)  2c<2a, 

If  X  and  1/  are  the  coordinates  of  the  point  P,  it  is  seen 
from  the.  figure  that 

(3)  r,^  =  ^e  +  xy  +  f; 

(4)  r^^  =  C<^-xy  +  y\ 

If  these  values  of  r^  and  r^  be  substituted  in  equation  (1), 
the  equation  showing  the  relation  between  x  and  3/  is 
obtained ;  that  is,  the  equation  of  the  ellipse.  This  is  best 
done  as  follows : 

In  order  to  avoid  calculations  with  radical  signs,  we  raise 
equation  (1)  to  the  second  power,  obtaining  : 

r^  +  ^2^  —  4  6fc2  —  _  2  r^^. 
By  squaring  the  latter  equation,  we  find 

(5)  (t^  +  r^y'  -  8  a\r^  +  r/)  +  16  a^  =  4  r^Vg^, 
or,  transposing  and  rearranging, 

(6)  (r^  -  r^y^  -  8  a\r:^  +  r^)  +  16  a*  =  0. 
From  equations  (3)  and  (4), 

r^JrT^J=2{x^^y^^c'), 
r^  —  r^  =  ^  ex  '^ 
and  by  substitution  in  (6), 

16  A2  -  16  a?(x^  +  ^2  +  ^2)  +  16  a*  =  0,  • 

or  (7)  x^  (cfi  -  c?2)  +  ay  =  a2  (^2  -  ^2). 


44  CALCULUS  [Ch.  I. 

Dividing  both  sides  hy  a^(^a^  —  c^),  we  have 

\i  B^  is  a  point  of  the  axis  of  ordinates  such  that 

and  if  we  designate  OB^  by  5,  we  obtain  from  the  right 
triangle  B^F^  0, 

(9)  a^-c^^h^i 

and  by  substitution  our  equation  passes  into  the  form 

(10)  ^  +  ?^=»- 

a"      6" 

This  is  the  equation  of  the  ellipse.  The  ratio  -  is  known  as 
the  eccentricity  of  the  ellipse. 

Art.  15.  The  form  of  the  ellipse.  We  endeavor  next  to 
get  an  idea  of  the  form  of  the  ellipse.  Giving  x  any  definite 
value,  as,  for  example,  x=  OQ,  the  corresponding  values  of  y 
are  found  from  equation  (10)  to  be 

y-- 

For  every  value  of  x  there  are  two  values  of  y  that  differ 
only  in  sign,  and  hence  determine  two  points  of  the  locus, 
P'  and  P",  lying  symmetrically  with  reference  to  the  axis 
of  abscissa).  The  x-axis  is  accordingly  an  axis  of  symmetry 
for  the  ellipse  ;  since  the  equation  of  the  ellipse  has  the  same 
form  for  both  y  and  x^  it  follows  that  the  y-axis  is  also  an 
axis  of  symmetry  for  the  ellipse.  The  axes  divide  the  ellipse 
into  four  congruent  quadrants;  the  discussion  of  one  of 
these  quadrants  will  suffice. 


14-16.]      THE  ELEMENTS   OF  ANALYTIC  GEOMETRY  46 

We  ask,  therefore,  how  will  t/  change  when  x  increases 
from  zero  ?  If  a;  =  0,  then  y  =  ±b  \  that  is,  the  point  B^^ 
as  well  as  B^^  which  is  symmetrical  with  it,  are  points  of 

the  ellipse.     If  x  increases,  1 (and  with  it  ?/)  becomes 

smaller  and  smaller,  until,  when  x  =  a^  both  1 and  y 

become  equal  to  zero  ;  the  values  x~a^  y  —  ^  determine 
a  point  of  intersection  of  the  ellipse  with  the  a:-axis ;  the 
symmetrically  located   point  A^  is  likewise  a  point  of  the 

ellipse.     If  x  still  continues  to  increase,  -i:  >  1,  and   1 

becomes  negative ;  and  there  is,  therefore,  no  real  value  of 
y  corresponding  to  a  value  of  x>  a.  Consequently,  the 
points  of  the  ellipse  all  lie  within  the  strip  bounded  by  two 
lines  drawn  parallel  to  the  axis  of  ordinates  and  through 
the  points  A^  and  A^.  In  a  similar  manner  it  follows  that 
they  also  all  lie  within  a  strip,  which  two  lines  form  drawn 
through  B-^  and  B^  and  parallel  to  the  axis  of  abscissae. 
Hence  the  ellipse  lies  within  a  rectangle  with  sides  2  a  and 
2  6,  as  shown  in  the  figure. 

A^A^  is  termed  the  major 
axis,  and  B^B^^  the  minor  axis 
of  the  ellipse.  The  lengths 
of  these  axes  are  2  a  and  2  h ; 
a  and  h  themselves  are  the 
semiaxes.  The  points  A^^  A^, 
J?i,  B^  are  named  vertices,  and 
the  points  J\  and  F^,  foci. 


Fig.  22. 


Art.  16.    Problems  concern- 
ing the  ellipse.    Geometrically 

defined,  the  tangent  to  a  curve  is  that  straight  line  which 
is  the  limiting  position  which  a  secant   line  approaches  as 


46  CALCULUS  [Ch.  I. 

two  consecutive  points  of  intersection  approach  coincidence. 
Thus  (Fig.  22),  as  the  secant  line  moves  parallel  to  itself,  the 
two  points  of  intersection  move  closer  and  closer  together, 
and  finally  coincide  at  P.     The  line  APB  may  therefore 

be  regarded  as  a  special  case  of  a 
secant  line  in  which  two  intersec- 
tions coincide.  The  same  result  may 
also  be  obtained  (Fig.  23)  by  keep- 
ing one  of  the  points  of  intersection 
fixed,  and  revolving  the  line  about 
it  as  a  pivot  until  a  second  point  of 
intersection  comes  to  coincidence 
with  it.  It  follows  from  what  we 
have  said  that  to  find  the  equation 
of  a  tangent  to  a  curve  whose  equa- 
tion we  know,  we  must  find  the 
equation  of   a  secant  line  in  which 

Fig.  23.  .  „   .  -        i  i 

two  points  ot  intersection  have  been 
made  to  coincide.  We  take  up  a  few  problems  which  will 
sufficiently  explain  the  method. 

I.  To  find  the  equation  of  the  secant  line  through  the  points 
(j^iVi)  ^^^  (^2^2)  ^^  ^^^  ellipse^ 

We  have  seen  that,  regarding  these  points  as  any  points, 
without  reference  to  the  ellipse,  the  equation  of  the  straight 
line  through  them  is  (p.  33), 

y^-Vx  ^2-^1 


16.]  THE  ELEMENTS   OF  ANALYTIC  GEOMETRY  47 

But  if  the  points  lie  on  the  ellipse,  their  coordinates  must 
satisfy  the  relations 


«2  +    52          ^  '       ^2  +   52   -  -^ 

subtract 

Aug  and  transposing, 

(2) 

52       -            a2 

Multiplying  equations  (1)  and  (2),  member  by  member, 
we  have 

62  -^  «2 


(3) 


^^  a2     ^  +  ^52—^- ^2 + p 

a^       52        ^2         62 

~     "^    a2  "^    62  * 

This  is  the  equation  of  the  straight  line  through  two  points 
of  the  ellipse.  If  these  two  points  are  brought  to  coinci- 
dence, the  secant  line  becomes  the  tangent.  Letting  (x^^  y^ 
coincide  with  (a;^,  y^),  we  have 

or   (5)  ■  ^  +  ^  =  1. 

This  is,  therefore,  the  equation  of  the  tangent  to  the 
ellipse  at  the  point  x^y^  on  it. 

A  simpler,  and  at  the  same  time  more  general  method  of 
determining  the  tangent  will  be  established  in  the  Calculus, 


48  CALCULUS  [Ch.  I. 

II.  To  find  the  condition  that  a  given  straight  line  may 
touch  the  ellipse. 

Let  the  equation  of  the  straight  line  be 

y  =  mx  +  c. 

Regarded  geometrically,  the  straight  line  will,  in  general, 
intersect  the  ellipse  in  two  distinct  points,  and  in  the  par- 
ticular cases  in  which  these  points  are  coincident,  the  given 
line  will  be  tangent  to  the  ellipse.  We  shall  first  find  the 
intersections,  and  then  determine  under  what  conditions 
they  are  coincident.  The  coordinates  of  the  points  of  inter- 
section must  satisfy  both  the  equations 

y=mx  +  G  and  —  +  f-  =  1. 

Hence  for  these  points  we  have 

x^      (mx  4-  c')^  _  1 

-2+  P  -'• 

or  (6)       (6^  -f  a^m^)  x^  +  2  a^cmx  +  a^c^  —  a^U^  =  0. 

The  roots  of  this  equation  will  be  the  values  of  the 
abscissae  of  the  points  of  intersection,  and  for  each  abscissa 
the  equation  y  =  mx  +  c  will  determine  one  ordinate.  There 
are,  then,  two  points  of  intersection,  and  if  the  roots  of  the 
equation  in  x  are  equal,  these  points  will  have  the  same 
coordinates;  i.e.  will  coincide.  But  the  condition  that  the 
equation  (6)  shall  have  equal  roots  is  * 

(7)  (2  a'^cmy  -  4  (62  +  a^m^^  (a^c^  -  aW)  =  0, 

*  Formula  73,  Appendix. 


16.]  THE  ELEMENTS   OF  ANALYTIC  GEOMETRY  49 

which  reduces  to        a^m?  -\-  P  —  c^  =  0,*  or 


(8)  c  =  ±  Va2m2  +  b^. 

Whenever  e  and  m  are  so  related  that  the  relation  just 
written  is  satisfied,  then,  and  only  then,  the  line 

y  =  mx  +  c 

is  tangent  to  the  ellipse.     In  other  words,  the  line 

(9)  y  =  mx  ±  va-^m"  +  b" 

is  tangent  to  the  ellipse,  no  matter  what  value  m  may  have, 
and  will  represent  all  possible  tangents  by  giving  m  all 
possible  values.  It  is  the  general  equation  of  the  tangent 
to  the  ellipse. 

EXAMPLES 

1.    To  find  the  tangent  from  the  point  (3,  2)  to  the  ellipse 

^  +  1/1=1 
5      11        * 

Here  a^  =  5,b^  =  11,  and  hence,  by  equation  (9)  above, 

y  =  mx  rh  v^5  m^  +  11 

is  tangent  to  the  ellipse  for  all  values  of  m. 

We  have  yet  to  choose  rn  so  that  the  line  passes  through  the  point 
(3,  2).     Since  this  is  to  be  the  case,  we  must  have 

2  =  3  m  ±  V5  m^  +  11. 
Solving  this  equation,  we  find 

m  =  I  or   —  ^. 

Substituting  this  value  of  m  above,  we  have 

^22 

*  To  reach  this  form,  we  divide  by  a^,  thus  making  the  assumption  that 
a  is  not  zero.     If  a  were  zero,  the  ellipse  would  consist  of  a  single  point. 


50  CALCULUS  [Ch.  I. 

Of  these,  only  the  line  y  =1^ -VL 

passes  through  the  point  (3,  2)  *    Similarly,  the  value  m  =  -  ^  gives 

X  ,  7 
^=-2+2 

as  the  second  tangent  from  the  point  (3,  2)  to  the  ellipse. 

2.  What  is  the  equation  of  the  tangent  to  the  ellipse 

at  the  point  (1,  —  f )  ?     Here 

x^  =  l,  y,  =  -  f,  a2  =  4,  &2  =,  3^ 
and  we  have  as  the  equation  of  the  tangent, 

^-^=  1 
4     2* 

3.  Find  the  equation  of  the  tangents  to  the  ellipse 

ar2     2/2  _ 

which  make  an  angle  of  ^b°  with  the  x-axis. 

In  this  case  m  =  tan  45°  =  1,  a^  _  2,  52  _  7^  ^nd  the  lines  are  2/  =  a:  + 
V2  +  7,  or  ?/  =  a:  +  3  and  ?/  =  x  —  3  are  the  two  tangents  which  make  an 
angle  of  45°  with  the  a:-axis. 

Art.  17.  The  auxiliary  circle;  the  directrix;  the  eccen- 
tricity. 

I.  Let  us  consider  an  ellipse  (Fig.  24)  with  the  semi-axes 
a  and  5,  and  a  circle  having  the  major  axis  of  the  ellipse  as 

*  The  general  equation  of  the  tangent  to  the  ellipse  shows  that  there  are 
always  two  tangents  having  the  same  direction  (as  determined  by  ?7i),  which 
is  plain  geometrically.  In  our  problem,  We  determine  the  directions  of  the 
particular  tangents  which  pass  through  the  given  point.  Of  course,  only 
one  of  the  two  tangents,  having  the  direction  found,  will  pass  through  the 
point. 


16-17.]      THE  ELEMENTS  OF  ANALYTIC  GEOMETRY 


51 


diameter ;  and  let  P  and  P'  be  points  of  the  ellipse  and  of 
the  circle  lying  on  the  same  perpendicular  to  the  a;-axis. 
These  points  have  the  same  abscissa  a;,  but  different  ordi- 
nates,  which  we  designate  by  ^  and  y'.  The  equation  of 
the  ellipse  is  then  satisfied  by  x  and  ?/, 
and  that  of  the  circle  by  x  and  y' ;  i.e. 


f2 


or   (1) 


and     (2) 


-!  +  <'  =  !, 


+  ^  =  1. 


Fig.  24. 


Whence  by  subtracting  and  transposing, 


a2 

1  = 

y 

a 

P'Q 
PQ 

a 

(3) 


or, 


accordingly  the  corresponding  ordinates  of  the  ellipse  and  the 
circle  are  in  a  constant  ratio.  This  circle  is  called  the  auxil- 
iary circle  of  the  ellipse. 

II.    To  determine  the  distances  of  any  point  P  of  the  ellipse 
from  the  foci  F^  and  F^. 

According  to  p.  43,  the  distance  PF^  or  r^  is :     * 

(4)  r^^  =  Qc-xy  +  y\ 

The  value  of  ^^  as  deduced  from  the  equation  of  the  ellipse  is 


y 


2  =  52  - 


^^2 


■X". 


52 

CALCULUS 

But  (p.  44). 

p^c'-  =  a^; 

further, 

hence  we  get 

r^  =  a^-2cx^ 

or  (5) 

2     f        ^    ^^ 
rJ=[a x] 

In  the  same  way 

it  is  found  that 

[Ch.  I 


(^ 


(6) 


(^c-^xf  +  y'  =  {a+~ 


Fig.  25. 

Therefore  the  values  of  the  distances  PF^  and  PF^  are 


(7) 
and 


^  a 


r^  =  a X. 

2  a 


From  this  an  important  result  may  be  obtained. 

We  put  —  =  l-> 

c 

that  is,  c:  a  =  a:l, 

I  is  a  distance  defined  by  a  and  c  which  may  be  constructed 
as  the  hypotenuse  of  a  right  triangle  in  which  a  is  one  side 
and  c  is  its  projection  upon  the  hypotenuse,  just  as  is  shown 


17.]  THE  ELEMENTS   OF  ANALYTIC  GEOMETRY  53 

by  the  figure.  We  lay  off  OD2  equal  to  ?,  and  draw  through 
i>2  ^  liii6  ^2  parallel  to  the  axis  of  ordinates ;  if  a  perpen- 
dicular from  P  be  dropped  on  this  line,  it  is  easily  seen  that 

(8)  Pl)^  =  l-x  =  --x. 

Writing  7*2  in  the  form 

and  substituting  from  (8)  we  have 

The  straight  line  d^  is  called  the  directrix  of  the  ellipse ; 
the  foregoing  equation  states  in  regard  to  it  that  the  ratio 
of  the  distances'  of  any  point  of  an  ellipse  from  the  focus  and 

from  the  directrix  has  the  fixed  value   -• 

As  a  consequence  of  the  symmetry  of  the  ellipse  there 
must  be  another  directrix  d^  belonging  to  the  focus  F^^  the 
distance  of  which  from  0  is  also  equal  to  Z,  and  for  which 
the  same  law  holds  true. 

The    ratio    -   has   already  been   defined    (p.   44)   as  the 


eccentricity  of  the  ellipse.  Since  c  =  V^^TTp^  and  hence 
c  <a^  the  eccentricity  of  the  ellipse  is  always  less  than 
unity.     As  2 

02)2  =  ?  =  -, 
^  c 

we   can   readily  write   the   equation  of   the   directrix,  viz. 

x  —  ~     or  x= ,  for  the  other  directrix  . 

c    \  c  J 


54  CALCULUS  .  [Ch.  I. 

EXERCISES  VII 

Note.  Throughout  this  set  of  exercises,  the  axes  of  the  ellipse  are 
taken  to  be  the  coordinate  axes. 

1.  Construct  the  equations  of  the  ellipses  having  as  semi-major  and 
semi-minor  axes  respectively  the  following  pairs  of  values  : 

(i.)3,2;  (ii.)4,  1;  (iii.)6,2;  (iv.)  5,  5. 

2.  Find  the  coordinates  of  the  foci  of  the  ellipses  in  1. 

3.  Find  the  equations  of  ellipses  which  have  respectively  the  follow- 
ing semi-major  axes  and  foci : 

(i.)  3,(2,0)  (V.)  3,(0,0); 

(ii.)  5,(4,0);  (vi.)  2,  (-1,0); 

(iii.)  6,  (-3,0);  (vii.)  h,{-k,Q). 

(iv.)  7,  (-1,0); 

4.  Find  the  equations  of  ellipses  having  respectively  the  first  of. 
each  of  the  following  pairs  of  points  as  focus,  and  passing  through  the 
second : 

(i.)    (2,0),  (2,3);  (iii.)   (5,0),  (-2,  -5); 

(ii.)   ( -  3,  0),  (3,  -  -V)  ;  (iv.)   (/,  0),  (c,  d). 

5.  Find  the  equations  of  ellipses,  referred  to  their  axes  as  axes  of  co- 
ordinates, and  passing  respectively  through  the  following  pairs  of  points 
(cf.  method  of  p.  33,  p]qs.  5  .  .  8)  : 

(i.)   (3,1),  (1,5);  (iii.)   (-2,6),  (4,  -2); 

(ii.)   (4,  -  2),  (-  3,  -  3);        (iv.)   (1,  3),  (6,  -  1). 

6.  Find  the  equations  of  the  tangent  from  the  j)oint  (—  1,  6)  to  the 

ellipse 

—  4-  ^y'^  —  1 
36       sT" 

7.  Find  the  equation  of  the  tangent  to  the  ellipse 

2x2  +  82/2  =  1, 
at  the  point  (—  |,  i). 

8.  To  which  of  the  following  ellipses,  if  any,  is  the  line  a:  =  11  —  3^ 
tangent? 

(i.)  3x2  +  2/2^4; 

(ii.)  2x2  +  5/ =  1; 

(iii.)  8x2  + 16?/2  =  176; 

(iv.)  ^'  +  J^  =  1. 

^     ^  25      100 


17-18.J      THE  ELEMENTS   OF  ANALYTIC  GEOMETRY 


65 


9.  Find  the  general  equation,  representing  all  tangents  to  each  of 
the  ellipses  of  8. 

10.  P'ind  the  equations  of  each  tangent  to  every  ellipse  of  8  which 
makes  equal  intercepts  on  the  axes. 

11.  Determine  the  eccentricity  and  the  equation  of  the  directrix  of 
each  of  the  following  ellipses  : 


o 


(ii.) 

(iii.)  5a:2  + 10?/2=:  1. 

12.  Find  the  relation  between  the  corresponding  abscissae  of  points 
on  the  ellipse  and  on  the  circle  whose  diameter  is  the  minor  axis  of  the 
ellipse. 

Art.  18.  The  equation  of  the  hyperbola.  We  define  the 
hyperbola  as  the  locus  of  a  point  which  moves  so  that  the  differ- 
ence of  its  distances  from  two  fixed  points  is  constant.  Let  the 
value  of  this  constant  difference  he  2  a. 


Fig.  26. 


We  take  (Fig.  26)  the  line  connecting  the  fixed  points 
Fi  and  F^  as  the  axis  of  abscissse,  and  the  perpendicular  at 
its  middle  point  0  as  the  axis  of  ordinates.     We  designate 


56  CALCULUS  [Ch.  I. 

by  2  c  the  distance  F^F^^  and  by  r^  and  r^^  respectively,  the 
distances  from  F^  and  F^  of  the  point  P  (lying  on  the  hyper- 
bola), so  that,  by  definition, 

(1)  r^  -  ^2  =  2  a. 

Since  in  every  triangle  the  difference  of  two  sides  is  less 
than  the  third  side,  it  follows  that 

(2)  2a<2c. 

If  the  coordinates  of  P  be  a:  and  y,  it  is  seen  from  the  figure 
that 

(3)  ri2  =  (:r  +  0'  +  /; 

(4)  ^^2_(^_,)2  +  ^2. 

By  squaring  (1)  and  transposing  we  get 

r^  +  ^2^  —  4  a^  =  2  r^r^, 
and  by  squaring  again, 

(5)  (r^  +  ^2^)2  -  8  a\r^  +  r^)  +  16  a*  =  4  r^r^^. 

Equations  (3),  (4),  (5)  are  identical  with  (3),  (4),  (5)  on 
p.  43,  and  by  proceeding  just  as  we  did  there,  we  obtain 

(6)  ^'^  -^      '^—  -  ■• 


In  the  case  of  the  hyperbola,  however,  a  <  c,  as  seen  above 
therefore  a^  —  c^  is  negative.     Putting 

(7)  a?-c^  =  -  h\ 

we  obtain  the  equation  of  the  hyperbola  in  its  final  form : 


18-20.]      THE  ELEMENTS   OF  ANALYTIC  GEOMETRY  57 

Art.  19.  The  form  of  the  hyperbola.  As  on  p.  44,  here 
also  both  the  axes  of  coordinates  are  axes  of  symmetry.  To 
show  the  course  of  the  hyperbola  in  the  first  quadrant,  the 
equation  may  be  written 

it  is  apparent  that  the  right  member  is  negative  so  long 
as  x<  a\  hence  for  all  such  values  of  x  the  corresponding 
values  of  y  are  imaginary,  i.e.  no  corresponding  geometric 
point  of  the  hyperbola  exists.  No  part  of  the  hyperbola  lies 
in  the  strip  between  the  perpendiculars  to  the  a;-axis  at  x  =  a 

and  x  =  —  a.     If 

X  =  a^ 

the  right  member  is  zero,  and  the  pair  of  values 

A    x  =  a, 

and  -^^    ^  =  0 

determines  the  point  A^  of  the  positive  a;-axis  through  which 
the  hyperbola  passes  ;  the  point  A^  of  the  negative  a;-axis 
(^OA^  =  —  a)  also  belongs  to  the  hyperbola.  If  x  increases 
still  more,  i.e.  if  x  has  any  value  greater  than  a,  y  is  always 
real  and  increases  continuously  with  x ;  the  hyperbola  ex- 
tends to  an  unlimited  distance  in  each  quadrant  as  is  seen 
in  the  figure. 

The  line  A^A^  is  termed  the  real  axis  of  the  hyperbola, 
and  the  axis  of  ^  —  from  analogy  —  the  imaginary  axis. 
The  points  A^  and  A^  are  called  vertices  ;  a  is  the  real  and  h 
the  imaginary  semi-axis;  F^  and  F^  are  again  named  foci. 

Art.  20.  The  directrix  of  the  hyperbola.  To  calculate  for 
the  hyperbola  the  distances  of  any  of  its  points  from  the  foci. 


58  CALCULUS  [Ch.  I 

According  to  p.  56  we  have  for  the  distance  PF^  or  r^ 

(1)  r^^=(^x-\-cy  +  y\ 

'  As  in  the  analogous  case  for  the  ellipse  (II.,  p.  51),  we 
obtain 

(2)  /•,2  =  g:.+  aJ, 

whence 

(3)  r^  =  ^x  +  a, 

a 

and  similarly, 


c 


(4)  r^  =  -x-a, 

a 

The  value  of  r^  may  also  be  written  thus  : 

(5)  ,,  =  .(.-5. 
We  put 

(6)  ^  =  1, 

c 

i.e.  e  :  a  =  a  :  l^ 

I  being  defined  by  a  and  (?,  just  as  the  analogous  distance  on 
p.  52 ;  here,  however,  a>l,  since  c>  a.  We  lay  off  OD^ 
(Fig.  26)  equal  to  Z,  draw  through  B^  the  line  d^  parallel  to 
the  ^-axis,  and  let  fall  upon  it  from  P  the  perpendicular 
PDo ;  we  then  have,  as  before. 


PD,- 

=  x-l  = 

X 

c 

and 

by  substituting  this  in  (5)  we 

find 

(7) 

PF,_ 

e 

As  in  the  case  of  the  ellipse  the  straight  line  d^  is  called  the 
directrix  of  the  hyperbola;  the  distance  of  any  poi7it  of  the 
hyperbola  from  the  focus  and  from  the  directrix  are  in  the 


20-21.]      THE  ELEMENTS   OF  ANALYTIC  GEOMETRY 


59 


constant  ratio  c :  a,  A  directrix  d^  belongs  to  the  focus  J\, 
for  which  the  same  law  holds.  The  ratio  c  :  a  is  called  the 
eccentricity  of  the  hyperbola. 

Art.  21.   The  equilateral  hyperbola  and   its  asymptotes. 

If  it  be  assumed  that  the  axes  2  a  and  2  5  of  an  ellipse  are 
equal  to  each  other,  the  ellipse  passes  into  the  circle ;  the 
circle  is  therefore  the  simplest  case  of  the  ellipse,  li  a  =  h 
in  the  equation  of  the  hyperbola,  a  remarkably  simple  hyper- 
bola is  obtained,  which  is  called  the  equilateral  hyperbola. 
Its  equation  is 

(1)  ^-^=i> 


or  a;2  _  ^2  _  ^2^ 

We  term  the  lines  bisecting  the  angles  between  the  coordi- 
nate axes  the  asymptotes  of  this  hyperbola,  and  propose : 

To  find  the  equation  of  the  equilateral  hyperbola  when  its 
asymptotes  are  taken  as  axes. 

As  an  aid  to  the  solution  of 
the  problem  we  present  the  fol- 
lowing preliminary  consider- 
ations. We  take  any  two 
straight  lines,  passing  through 
0  (Fig.  27)  and  at  right  angles 
with  each  other,  as  the  axes  of  a 
new  system  of  coordinates.     Let 

the  coordinates  of  any  point  P  referred  to  them  be  f  and  r;. 
We  draw  PQ  perpendicular  to  the  axis  of  x  and  PR  perpen- 
dicular to  the  axis  of  f,  so  that  RPQ  =  ROQ^(a  and 

(2)  OQ^x,     PQ  =  y. 

0R  =  ^,     PR  =  rj, 


■\ 

Y 

1 

3 

\ 

..^ 

^ 

y 

^^0 

k 

X        ( 

2  T 

Fig.  27. 


60 


CALCULUS 


[Ch.  I. 


and  further  draw  RT  perpendicular  and  RS  parallel  to  tlie 
axis  of  X.     It  then  follows  that 

x=  0Q=  OT-TQ=  OT-RS, 
y=PQ^PS+  QS  =  PS  +  RT, 

But  from  the  triangles  ORT  smd  PRS, 

0T=^  cos  a, 

RS  =  7]  sin  a, 

RT=^^ma, 

PS  =  7]  COS  a, 

and  hence, 

(3)  x=  ^  cos  a  —  Tj  sin  a, 

y  =  i  sin  a  -{-  7]  cos  a  ; 

and  these  are  the  equations  which  show  how  the  coordinates 
of  a  point  P  referred  to  one  system  of  axes  are  related  to  its 
coordinates  in  the  other  system. 

Applying    this    to    the    asymptotes   taken    as    new   axes 
(Fig.  28),  we  see  that  a  =  -  45°  (p.  22),  and  therefore 

cos  a  =  V|-, 
and       sin  ot  =  —  V|- ; 
we  accordingly  obtain  for  this 
special  case  the  equations 

(4)  ^  =  |VI  +  77VI, 

2/  =  -  f  VI  -f  7;Vr 

whence, 

(5)  x-}/=2^Vl 

Fig.  28.  X  +  1/  =  2  7/ Vj. 


21.]  THE  ELEMENTS   OF  ANALYTIC  GEOMETRY  61 

If  P  is  a  point  of  an  equilateral  hyperbola,  its  coordinates 
must  satisfy  the  equation 

^2  _  ^2  _  ^2^ 

which  we  can  also  write  in  the  form 

(6)  (x  +  y){x  -  y^  =  a\ 

By  introducing  the  previous  values,  we  get  as  the  equa- 
tion of  the  same  point  P,  referred  to  the  axes  f,  t;, 

(7)  nn^a\ 

and  since  the  coordinates  of  any  point  of  the  hyperbola 
satisfy  this  relation,  tM%  equation  is  the  equation  of  the  equi- 
lateral hyperbola  referred  to  its  asymptotes  as  coordinate  axes. 
We  now  see  that  Boyle's  Law  is  represented  graphically 
by  a  hyperbola;  for  if  we  substitute  p  for  f  and  v  for  ?;, 
and  put 

^^  _  1 

I  ~  ' 

the  equation  becomes 

pv  =  1, 

The  following  geometric   property  of  the   asymptotes  is 
interesting.      By  writing  equation  (7)  in  the  form 

^=2-x 

we  see  that  the  smaller  y  is  the  greater  is  x;  i.e,  the  hyper- 
bola approaches  nearer  and  nearer  the  axis  of  x  tlie  farther 
it  extends,  but  never  reaches  it,  no  matter  how  large  x  may 
become.  (An  abbreviated  form  of  this  statement  ofte^  used 
is  that  the  hyperbola  reaches  the  axis  of  x  only  if  x  =  cc. 
This  mode  of  abbreviation  will  be  discussed  in  tlie  next 
chapter.)     A  similar  statement  is  true  of  the  axis  of  y.     For 


62  CALCULUS  [Ch.  I. 

this  reason,  the  axes  of  x  and  y  are  called  asymptotes  *  of  the 

hyperbola;  the  hyperbola  approaches  nearer  and  nearer  to 

both  lines  the  farther  they  extend,  but  never  reaches  them. 

In  all  the  preceding  articles  the  axes  have  been  supposed 

to  be  at  right  angles  to  each  other.     It  is  possible  to  treat 

all  the  problems  which  we  have  hitherto  discussed  without 

making  this  assumption,  but  as  the  results  when  the  axes 

are  not  at  right  angles  with  each  other  are  of  much  less 

importance,  we  pass  them   by  with  this  mention.     In  the 

exercises  which  follow^   the    axes  are   always  supposed  to   he 

rectangular. 

EXERCISES   VIII 

1.  Show  (in  a  manner  analogous  to  that  explained  in  the  case  of  the 
ellipse)  that  the  equation  of  the  tangent  to  the  hyperbola,  at  the  point 
ajj^i  on  it,  is 

2.  Show  likewise  that 

y  =  mx  ±  ^a^mP'  —  W- 

is  tangent  to  the  hyperbola  for  all  values  of  m. 

3.  Using  the  results  of  the  previous  exercises,  find  the  equations  of: 
(i.)  The  tangent  to  the  hyperbola 

^  _  .^^  _  1 
T      12  " 
at  the  point  (4,  —  6). 

(ii.)  The  tangents  to  the  same  hyperbola  from  the  point  (—  1,  1). 
(iii.)  The  tangents  to  the  same  hyperbola  from  the  origin, 
(iv.)  The  tangents  from  the  point  (6,  2).     Interpret  this  result  geo- 
metrically. 

4.  Find  the  tangents  from  the  origin  to  the  hyperbola 

X^  —  y^  r=L  ^. 

5.  Find  the  equation  of  the  equilateral  hyperbola 

a;^  —  ^^  =  9 
referred  to  its  asymptotes  as  axes. 

*  Grk.  dav/xwTOJTos,  not  falling  together. 


Y 

jY 

id 

p 

R 

""~T- 

0 

iN 

5 

21-22.]      THE  ELEMENTS   OF  ANALYTIC  GEOMETRY  63 

Art.  22.  Transformation  of  coordinates.  We  solved  the 
problem  treated  in  the  last  article,  by  introducing  a  new 
system  of  coordinates.  The  introduction  of  new  systems 
of  coordinates  is  often  of  great  importance.  The  position  of 
the  axes  is  indeed  arbitrary,  but  it  is  readily  seen  that  there 
will  usually  be  for  every  curve,  or  geometric  construction, 
some  preferable  position.  Generally  it  cannot  be  deter- 
mined which  position  this  is, 
until  the  equation  of  the  curve 
when  referred  to  an  arbitrarily 

assumed    system    of    axes    has     , ;r__j.'.' v 

been  deduced.      We  must  ac- i___L_X 

cordingly     establish     formulae 
that   will    enable    us    to    pass  „      ^^ 

from  equations  referred  to  one 

system  of  coordinates  to  equations  referred  to  another  sys- 
tem. To  begin  with,  we  assume  the  axes  of  both  systems  to 
be  parallel  to  one  another.     For  example,  in  Fig.  29,  let 

(1)  X=0'R,     Y=FB, 

be  the  coordinates  of  the  point  P  referred  to  the  axes  O^X' 
and  O'Y'.     Furthermore,  let 

(2)  x=OQ,       y=PQ. 

be  the  coordinates  of  the  point  P  in  the  system  of  coordi- 
nates whose  origin  is  at  0 ;  finally  let 

(3)  a=ON,        h  =  0'N, 

be  the  coordinates  of  the  point  0'  in  the  latter  system. 
Then,  between  the  coordinates  a;,  y  and  X,  Y  of  the  point  P, 
we  have  the  equations 

(4)  X=x~a,   Y=y-hi 

these  equations  are  true  of  every  point. 
6 


64  CALCULUS  [Ch.  I. 

The  equation  of  the  circle  for  the  coordinates  JT,  Y  is 

(5)  X2  +  F2  =  7-2, 

r  being  the  radius ;  this  equation  is  true  for  every  point,  P, 
of  the  circle.  If  we  substitute  for  X  and  Y  their  values  as 
given  in  (4),  we  get 

(6)  (^_a)24-(^-5)2=r2; 

this  equation  is  satisfied  by  the  coordinates  x,  ?/,  of  any  point, 
P,  of  the  circle,  and  is  therefore  the  equation  of  the  circle 
with  the  new  system  of  axes.  We  obtained  the  same  equa- 
tion in  a  different  way  on  p.  18. 

If  we  take  new  axes  having  the  same  origin  as  the  origi- 
nal ones,  but  different  directions,  the  formulae  (3),  (p.  60), 
are  to  be  employed,  viz.  : 

(7)  x=  ^  cos  a  — 7]  sin  a;      i/ =  ^  sin  a -^  rj  cos  a  ; 

by  multiplying  them  by  cos  a  and  sin  a  respectively,  and 
adding,  and  also  performing  the  same  operations  with 
—  sin  a,   +  cos  a,*  we  find  ' 

(8)  ^  =  X  cos  a-\-  ^  sin  a;     rj  =  —  x  sin  «  -h  y  cos  a. 

The  coordinates  x,  y  and  f,  t)  can  therefore  be  expressed, 
each  in  terms  of  the  others,  in  exactly  the  same  way.  We 
can  pass  from  one  system  of  coordinates  to  any  other,  having 
a  new  origin  and  different  directions  of  axes,  by  carrying 
out  two  transformations  of  coordinates,  one  after  the  other. 
The  transformation  of  coordinates  is  of  very  great  utility. 
By  its  aid  it  can  be  proved  that  (^disregarding  some  .excep- 
tions) every  equation  of  the  second  degree  represents  an  ellipse, 

*  Formula  28,  Appendix. 


22.]  THE  ELEMENTS   OF  ANALYTIC  GEOMETRY  65 

a  hyperbola,  or  a  parabola.  This  is  done  by  transforming 
the  system  of  coordinates  so  that  the  most  general  equation 
of  the  second  degree,  viz.^ 

(9)  ax^  -\-2bxi/  +  cf-j-2dx-{-2  ey  +f=  0, 

passes  into  one  of  the  forms  which  we  have  already  found 
for  the  equations  of  the  ellipse,  the  hyperbola,  and  the 
parabola. 

EXERCISES   IX 

1.  Find  the  equation  of  the  circle  : 

x^  +  y^  =  r\ 

(i.)  If  the  new  origin  is  at  the  upper  end  of  the  vertical  diameter 
and  («)  the  new  axes  are  parallel  to  the  old ;  (h)  if  the  new  axes  make 
an  angle  of  45°  with  the  old. 

(ii.)  If  the  new  origin  is  at  the  right  end  of  the  horizontal  diameter, 
and  (a)  the  new  axes  are  parallel  to  the  old,  (h)  the  new  axes  make  an 
angle  of  60°  with  the  old. 

(iii.)  If  the  axes  are  the  tangent  at  the  lower  extremity  of  the  vertical 
diameter  and  the  tangent  at  the  left  extremity  of  the  horizontal  diameter. 

2.  Show  that  in  transforming  from  one  system  of  rectangular  coordi- 
nates to  any  other,  the  degree  of  the  equation  of  any  curve  is  not  altered. 

3.  Find  the  equation  of  the  ellipse  referred  to  the  major  axis  and 
a  tangent  at  the  left  vertex. 

4.  Find  the  equation  of  the  hyperbola  referred  to  its  real  axis  and 
the  tangent  at  the  right  vertex. 

5.  Find  the  equation  of  4  a:^  _  p  ^2  _  35^  referred  to  axes  having  the 
same  origin  as  the  old  axes  and  making  an  angle  of  —  60°  with  them. 

6.  Find  the  equation  of 

a:2  _  5  y.y  ^  y2j^  8  oT  -  20  ?/  +  1 5  =  0, 

for  new  axes  with  origin  at  (—4,  0)  and 
(i.)    parallel  to  the  original  axes ; 
(ii.)   making  an  angle  of  —  45°  with  the  original  axes. 


66  CALCULUS  [Cii.  I. 

7.  Find  the  equation  of 

36  x^  +  2ixy  +  29ij^-72x  +  126  y  +  81  =  0, 
referred  to  new  axes  with  origin  at  (2,  —  3)  and 
(i.)   parallel  to  the  original  axes ; 
(ii.)    making  with  original  axes  an  angle  whose  tangent  is  —  |. 

8.  Find  the  equation  of 

y^  —  4:y  —  5x  =  0, 

referred  to  parallel  axes  with  origin  at  (—4,  2). 

9.  Find  the  equation  of 

16x2  +  25?/2  +  32x-  lOOy  -  284  =  0, 
referred  to  parallel  axes  with  origin  at  (—  1,  2). 

Art.  23.  Van  der  Waals's  equation.  The  equations  of  the 
curves  thus  far  considered  have  been  of  only  the  first  or 
second  degree.  Although  it  is  beyond  the  scope  of  this  book 
to  discuss  curves  with  equations  of  higher  orders  than  the 
second,  yet  we  mention  at  least  one  example  of  such  curves. 

The  equation  for  Boyle's  Law  (p.  3)  does  not  hold  true 
when  the  pressure  upon  a  gas  exceeds  certain  limits.  For 
the  case  of  strongly  compressed  gases,  van  der  Waals  has 
proposed  a  celebrated  equation,  which  commonly  goes  by  his 
name ;    viz,  : 

(1)  (^  +  £)(,_5)=l;, 

in  it  a  and  h  are  positive  constants,  characteristic  of  the 
gas  under  consideration. 

By  taking  the  volume  v  and  the  pressure  p  as  coordinates, 
we  can  represent  graphically  *  the  law  formulated  in  equa- 
tion (1). 

*  Since  when  we  multiply  out,  a  term  (pv^)  appears,  which  is  of  the 
fourth  degree  in  p  and  v  together,  the  curve  is  said  to  be  of  the  fourth 
order  (or  a  quartic  curve). 


22-23,]         THE  ELEMENTS   OP  ANALYTIC  GEOMETRY  67 

We  now  discuss  this  law  briefly. 

I.  If  we  allow  the  mass  of  gas  to  occupy  a  large  volume, 
that  is,  if  we  make  v  very  large,  the  value  of  —  in  equation 

will  become  very  small,  in  practice,  inappreciable  ;  v  —  h^ 
likewise,  will  not  differ  appreciably  from  v,  so  that  we  have, 
approximately, 

(2)  pv=^l', 

iu  other  words,  in  the  case  of  highly  rarefied  gases  van  der 
Waals's  equation  passes  over  into  that  for  Boyle's  Law. 

II.  If,  however,  v  is  not  very  large,  that  is,  if  the  gas  is 
not  in  a  condition  of  considerable  rarefaction,  the  influence 
of  the  constants  a  and  h  becomes  appreciable. 

If  we  make  v  very  small  by  increasing  the  pressure  p^ 
equation  (1)  in  which  jt?,  v,  a,  h  are  all  positive,  shows  that 
V  will  approach  h  in  value  constantly,  until,  when  p  is  enor- 
mously great,  v  approximately  coincides  with  h.  The  con- 
stant h  is  the  limit  to  the  smallness  of  the  volume  whicli  the 
gaseous  mass  can  be  made  to  assume  through  an  increase  of 
pressure ;  according  to  the  above  equation,  a  smaller  volume 
than  h  is  impossible. 

The  condition  of  affairs  is  made  most  evident  by  a  graphic 
representation.     For  carbon  dioxide  (carbonic  acid  gas), 

a  =  0.00874;  J  =  0.0023; 
hence 

The  pressure  is  reckoned  in  atmospheres  ;  if  we  put 
p  =  1,  we  obtain  from  the  above  equation  a  value  for  the 
volume,  which  is  easily  found  to  be  0.9936  ;  (i.e.  the  unit 
of  volume  is  a  little  larger  than  that  which  the  mass  of  gas 


68 


CALCULUS 


[Ch.  I. 


under  consideration  occupies  when  subject  to  a  pressure  of 
one  atmosphere). 

The  values  of  p  and  v,  given  in  the  following  table,  de- 
termine, when  plotted,  the  curve  shown  in  Fig.  30. 


T 

1) 

V 

p 

0.1 

9.4 

0.008 

38.8 

0.05 

17.5 

0.005 

20.9 

0.015 

39.9 

0.004 

42.0 

0.01 

42.6 

0.003 

45.7 

P 

~^ 

^ 

~~ 

~ 

AA 

- 

- 

-- 

50 

^ 

/ 

y 

_ 

~ 

~" 

~~ 

■~ 

""■ 

-J 

0.01 


0.025 
Fig.  30. 


The  equation 


'^  + 0,00874^^  ^_Q_0(,28  1=1 


0.05    V 


■A" 

is  true  only  when  the  temperature  is  0°  ;  for  carbon  dioxide 
at  the  temperature  ^,  van  der  Waals  gives  the  equation 


(;,  +  0.0m4j(^,_  0.0023)  = 


1  + 


273 


23-24.]         THE  ELEMENTS   OF  ANALYTIC  GEOMETRY 


69 


13.1 


if  we  assign  to  t  in  this  equation  different  values  (as  13.1, 
21.5,  etc.),  we  can  draw,  in  a  way  similar  to  that  above,  a 
curve  corresponding  to  p 
each  value  of  t.  This 
group  of  curves  (Fig. 
31)  gives  us  a  clear 
view  of  the  behavior 
of  carbon  dioxide  under 
the  most  various  con- 
ditions of  pressure,  vol- 
ume, and  temperature. 
By  its  consideration  van 
der  Waals  was  enabled 
to  draw  very  far-reach- 
ing conclusions  about 
the  behavior  of  matter  ( 
in  a  state  of  consider- 
able condensation,  both  in  the  gaseous  and  liquid  condition. 

Art.  24.  Polar  coordinates.  The  method  of  determining 
points  in  a  plane  by  means  of  the  coordinates  which  were 
defined  (pp.  7-10),  is  not  the  only  method  possible.  On  the 
contrary,  there  may  be  a  countless  number  of  such  methods, 
one  other  of  which  is  of  sufficient  importance  to  require 
mention  here. 

If  we  conceive  a  number  of  circles  to  be  drawn  around  0 
(Fig.  32),  and  if  we  draw  through  0  any  number  of  straight 
lines,  a  series  of  points  of  intersection  is  obtained,  each  of 
which  points,  as,  for  instance,  P^  or  P^,  has  its  position 
determined,  when  we  know  its  distance  r^  or  r^  from  0, 
and  the  angle  <^j  or  (f)^,  which  the  line  P^O  or  P^O  makes 
with  a  fixed  axis  OX.     As  on  p.  9,  here  also  we  find  that 


Fig.  31. 


70 


CALCULUS 


[Ch.  I. 


any  point  P  corresponds  to  a  pair  of  numbers ;   that  is,  to 
the  length  of  r  and  the  magnitude  of  (/>,  and,  conversely,  if  a 

pair  of  numbers,  as  r  and  <^,  is 
given,  one  point  P  can  always  be 
found  whose  position  in  the  plane 
is  defined  by  r  and  <^.  The  quan- 
tities r  and  <^  are  called  the  polar 
coordinates  of  the  point  P. 

To  determine  the  position  of  points  in 
the  plane  by  polar  coordinates,  we  make 
use  of  two  systems  of  lines ;  a  system  of 
Fig.  32.  concentric  circles  and  a  system  of  straight 

lines  passing  through  their  center.  In 
the  case  of  rectangular  coordinates,  we  made  use  of  two  systems  of 
straight  lines  respectively  parallel.  The  use  of  two  systems  of  lines  is, 
in  essence,  the  general  principle  underlying  every  type  of  coordinates 
employed  in  higher  mathematics. 

The  relations  between  polar  coordinates  and  rectangular 
coordinates,  for  which  OX  is  the  axis  of  abscissa?  and  0  the 
origin,  are  seen  from  a  consideration  of  triangle  OPQ 
(Fig.  32)  to  be 

(1)  x  =  r  cos  </),  y  =  r  sin  <^; 


(2) 


x^  -\-  y^  =  r^. 


By  means  of  these  equations  it  is  possible  to  calculate  the 
rectangular  coordinates  when  the  polar  coordinates  are 
known,  and  vice  versa,  the  polar  coordinates,  when  the  rectan- 
gular coordinates  are  given. 

Art.  25.  The  equations  of  the  ellipse,  the  parabola,  and 
the  hyperbola  in  polar  coordinates.  In  Arts.  14,  5,  and  20 
it  has  been  shoAvn  that  the  ellipse,  the  parabola,  and  the 
hyperbola  are  each  the  locus  of  a  point,  which  moves  so  that  its 


24-25.]        THE  ELEMENTS  OF  ANALYTIC  GEOMETRY  71 

distances  from  a  fixed  point  (^focus)  and  a  fixed  line  (direc- 
trix^ are  in  a  constant  ratio  (eccentricity~) ,  For  the  ellipse 
this  ratio  is  less  than  1  (since  <?  <  a),  for  the  hyperbola  it  is 
greater  than  1  (since  (?>«),  and  for  the  parabola  it  is  equal 
to  1  (since  both  distances  are  equal  in  the  parabola).  We 
deduced  this  property  from  certain  definitions  of  these 
curves.  We  might,  however,  have  set  out  with  it  as  defini- 
tion^ in  which  case  we  should  have  deduced  the  previous 
definitions  as  properties  of  the  curves.  This  would,  indeed, 
have  been  a  more  general  treatment,  since  the  three  curves 
would  have  been  comprised  under  one  definition,  and  it 
would  have  appeared  from  the  outset  that  the  three  curves 
are  all  varieties  of  one  general  type.  In  the  course  of  our 
study  of  the  curves,  we  have  seen  this  in  connection  with 
the  eccentricity,  and  also  in  connection  with  the  degree  of 
the  equations  of  the  curves,  all  of  the  equations  being  of  the 
second  degree,  and  together  constituting  the  totality  of  all 
curves  whose  equation  is  of  the  second  degree.  We  men- 
tion further  that  these  curves  are  all  varieties  of  the  plane 
sections  of  a  circular  cone,  whose  sides  are  produced  in 
both  directions  without  limit.  (The  section  is  an  ellipse,  a 
parabola,  or  an  hyperbola,  according  as  the  angle  between 
the  cutting  plane  and  the  axis  of  the  cone  is  greater  than, 
equal  to,  or  less  than,  the  angle  between  the  axis  and  the 
edge  of  the  cone.  For  this  reason  these  curves  are  often 
called  conic  section^ 

To  deduce  the  equations  of  the  conic  sections  in  polar 
coordinates,  we  use  as  definition  the  property  mentioned 
above.  We  designate  (Fig.  33)  the  distance  of  the  fixed 
point  F^  from  the  fixed  straight  line  d-^  by  jt?,  and  the  eccen- 
tricity by  ^,  and  proceed  to  derive  the  equation  of  all  three 
of  the  curves  by  one  process.     We  take  F^  as  the  origin  in 


72 


CALCULUS 


[Ch.  I. 


the  polar  coordinates,  and  the  perpendicular  F^L^  let  fall 
from  F^  on  d^,  as  the  axis ;  its  positive  half  shall  be  that 
which  does  not  intersect  the  right  line  d^ ;  we  have  then 


D 

P 

/ 

L 

P 

Fi 

Q 

di 

(1) 


FB 


e. 


Fig.  33. 


If  r  and  </>  be  the  polar  coordinates 
of  P,  and  P§  be  a  perpendicular 
from  F  on  the  axis,  then 

(2)    FF  =  LF^  +  F,Q  =  p  +  rcoscj>, 


and  by  substitution  we  obtain  the  required  equation  in  the 
form 


(3) 
whence 

J)  -{-  r  cos  (j> 

=  e 

(4) 

ep 

"*-   1- 

e  cos 

4> 

For  the 

ellipse  (pp. 

52-53) 

"-"a     ''- 

I  — 

c 

c 

so  that  its  equation  is 

W' 

(5) 

r  = 

a 

1- 

COS  (f) 

a 

For  the  hyperbola  (pp.  58-59) 


a  c  c  c 


25.]  THE  ELEMENTS  OF  ANALYTIC  GEOMETRY  73 

and  hence  its  equation  becomes  likewise 


(6) 


a 


1 cos  <f> 

a 


Since  in  the  parabola  ^  =  1,  its  equation  is  simply 


(7) 


2. 


1  —  cos  <f> 

where  p  is  the  parameter  of  the  parabola. 

The  fact  that  the  equations  of  the  ellipse,  the  parabola, 
and  the  hypeibola,  when  expressed  in  polar  coordinates, 
have  the  same  form,  is  of  great  importance  in  astronomy, 
particularly  in  the  determination  of  the  paths  of  comets. 
Every  comet  describes  an  ellipse,  a  parabola,  or  a  hyperbola, 
of  which  the  sun  is  a  focus.  Equation  (4)  is  the  equation 
of  the  comet's  path,  and  the  quantities  p  and  e  occurring 
in  it  are  to  be  determined  by  observations  on  various  posi- 
tions of  the  comet  in  the  heavens.  From  the  value  of  g,  it 
is  known  whether  the  comet  describes  an  ellipse  (e  <  1),  a 
parabola  (^  =  1),  or  an  hyperbola  (g  >  1).  In  the  first  case 
the  comet  moves  periodically  around  the  sun,  but  in  the 
other  cases  it  is  only  a  transient  guest  of  our  solar  system. 

Five  observations  are  sufficient  to  determine  the  orbit. 
Two  positions  of  the  comet,  together  with  that  of  the  sun, 
determine  the  plane  in  which  the  comet  moves,  and  the  three 
other  positions  are  needed  to  determine  the  three  quantities 
remaining  unknown,  viz.  the  direction  of  the  axis,  p  and  e,* 

*  A  sixth  observation  is  needed,  if  we  wish  to  determine  the  position  of 
the  comet  in  its  orbit. 


Fig.  34. 


74  CALCULUS  [Ch.  1. 

Art.  26.  The  Spiral  of  Archimedes.  We  shall  now  show 
how  curves,  whose  equations  in  rectangular  coordinates  are 
complicated,  can  be  represented  by  polar  coordinates  in  an 
extremely  simple  form.     Such  curves  are,  for  instance,  the 

spirals,  of  which  that  known  as  the 
spiral  of  Archimedes  is  an  example. 
Its  equation  is 

(1)  r=^a<f>. 

In  discussing  this    curve  we  shall 
find  it  most  convenient  not  to  meas- 
ure the  angle  <f)  in  degrees,  but  in 
circular  measurement.*     If  P^  and 
P^  (Fig.  34)  be  two  points  of  the 
spiral  which  belong  to  angles  <f)^  and  <^2'  (Offering  by  2  tt, 
they  lie  upon  one  and  the  same  straight  line  passing  through 
0.     The  equations 

(2)  rj  =  a(/)j,     and     r^  =  ac^g 


*  In  mathematical  computations,  angles  are  usually  measured  in  degrees  ; 
that  is,  the  unit  is  3 J^  of  the  angular  magnitude  about  a  point.  In  theoretic 
mathematics,  however,  it  has  been  found  advantageous  to  introduce  another 
unit  of  angular  measurement ;  namely,  that  in  which  the  unit  is  the  radian, 
or  the  angle  measured  by  an  arc  equal  in  length  to  the  radius.  This  system 
of  measurement,  known  as  circular  measure,  is  described  in  detail  in  works 
on  trigonometry. 

We  have  the  relation  360°  =  2  tt  radians,  which  enables  us  to  pass  from 
degrees  to  radians,  and  vice  versa.  No  symbol  has  been  generally  introduced 
for  the  unit  of  the  circular  measurement,  but  when  degrees  are  not  expressed, 
radians  are  understood  as  the  unit.     Thus  the  angles  ^,  -  are  the  angles 

which  contain  <p  and  -  radians  respectively  ;  ^  and  -  are  merely  numbers. 
^  Ji 

We  recall  also  that  in  trigonometry  we  have  extended  our  ideas  about 
angles  so  as  to  treat  of  angles  of  any  magnitude,  positive  as  well  as  negative, 
and  have  defined  the  trigonometric  functions  for  all  such  angles. 


26-27]         THE  ELEMENTS   OF  ANALYTIC  GEOMETRY  75 

hold  for  the  points  in  question,  and,  by  subtraction,  we  find 
(3)  ^2  -  ^1  =  ^(<^2  -  </>i)  =  2  air. 

Carrying  these  considerations  farther,  we  find  without 
trouble  that  there  is  a  countless  number  of  points  Pj,  P^^ 
Pg,  P^,  ••♦,  lying  on  every  straight  line  passing  through  0, 
which  belong  to  the  angles  <^j,  (/>2  =  (^^  +  2  tt,  c^g  =  <^j  +  4  tt, 
</>4=0i  +  6  7r,  •••;  the  distances  of  these  points  from  0  are 
given  by  the  equations  rg  =  r^  +  2  ^tt,  rg  =  r^  +  4  avr,  r^  =  r^ 
+  6  aTT,  •••.  Since  this  is  true  of  every  straight  line  radiat- 
ing from  0,  the  spiral  consists  of  a  boundless  number  of 
revolutions  which  wind  around  the  center,  always  keeping 
at  a  distance  2  air  from  one  another.  Inasmuch  as  for  </>  =  0, 
r=  0,  the  spiral  has  its  beginning  in  0. 

Art.  27.    Concerning    imaginary   points    and    lines.      We 

began  this  subject  by  establishing  a  correspondence  between  paii-s  of 
numbers  (called  coordinates)  and  points.  This  correspondence  was  such 
that  every  pair  of  real  numbers  determines  one,  and  only  one,  point 
of  the  plane,  and  vice  versa.  We  have  grown  accustomed  consequently 
instead  of  saying  "  the  point  whose  coordinates  are  a  and  h"  to  say  as 
abbreviation  "the  point  {n,  /:*)."  We  may  regard  the  pair  of  numbers 
itself  as  a  point  —  an  algebraic  "point,"  having  the  corresponding  geo- 
metric point  as  its  graphic  representation.  We  have  considered  also 
various  loci  or  graphs,  made  up  of  the  totality  of  all  points  whose 
coordinates  satisfy  a  certain  equation  given  in  each  case.  We  determine 
whether  or  not  a  certain  point  lies  on  a  given  curve  by  determining 
whether  or  not  its  coordinates  satisfy  the  equation  of  the  curve.  Point 
LIES  ON  CURVE  and  COORDINATES  SATISFY  EQUATION  are  perfectly 
equivalent.     We  now  extend  our  definitions. 

We  define  any  pair  of  numbers  (regarded  as  coordinates)  as  a 
"point";  i.e.  an  algebraic  point.  If  one  or  both  of  the  numbers  are 
imaginary,  we  call  the  point  an  imaginary  algebraic  point.  This  is  purely 
an  algebraic  conception  and  has  no  geometric  representation.  If  both 
numbers  are  real,  we  call  the  point  a  real  algebraic  point,  and  it  has  a 
geometric  representation.  We  now  say  any  point  (real  or  imaginary) 
lies  on  a  curve  if  the  coordinates  of  the  point  satisfy  the  equation  of 


76  CALCULUS  [Ch.  I. 

the  curve.     This  convention  enables  us  to  state  general  algebraic  theo- 
rems in  geometric  language.     Thus,  we  may  say : 

Every  straight  line  intersects  every  circle  in  two  points. 
This  means  neither  more  nor  less  than  : 

The  equations  x^  -\-  y"^  =  r^  and  ax  -\-  by  +  c  =  0  have  always  two  com- 
mon solutions  {real  or  imaginary). 

We  have  seen  that  the  equation  of  the  Jirst  degree  is  the  algebraic 
equivalent  of  the  straight  line.  This  was  proved  with  the  tacit  assumption 
that  the  coefficients  of  the  equation  are  all  real.  If  any  of  the  coeffi- 
cients become  imaginary,  the  equation  has  no  longer  any  geometric 
equivalent  in  our  system  of  coordinates.  However,  the  same  type  of 
relation  exists  between  the  two  numbers  of  every  pair  which  satisfies  the 
equation  {viz.,  the  ordinate  is  a  multiple  of  the  abscissa  plus  a  constant), 
and  therefore  we  now  say  that  every  equation  of  the  first  degree  is  the 
algebraic  equivalent  of  a  straight  line.  If  the  coefficients  are  all  real,  the 
straight  line  is  real,  and  has  a  graphic  representation.  If  any  of  the 
coefficients  are  imaginary,  the  straight  line  is  imaginary,  and  has  no 
graphic  representation.  Similarly,  we  speak  of  imaginary  curves  of  every 
species. 


CHAPTER   II 
CONCERNING  LIMITS 

Art.  1.    Constants,   variables,    and   limits.      A    variable 

quantity  is  one  which  may  assume  different  values  (usually 
a  boundless  number  of  them)  in  the  same  discussion.  The 
coordinates  x  and  ?/  of  a  point  on  a  curve  are  examples,  and 
the  law  according  to  which  they  vary  is  expressed  by  the 
equation  of  the  curve ;  in  the  case  of  the  straight  line  by 

y  =  mx  H-  b. 

A  constant  quantity  is  one  which  retains  the  same  value 
throughout  the  same  problem  or  discussion.  The  slope  of  a 
straight  line  and  its  intercept  on  the  axis  of  y  are  examples 
of  constants.  They  are  denoted  in  the  equation  above  by 
m  and  b. 

A  variable  quantity  which  is  considered  to  be  quite  arbi- 
trary, and  to  which  may  be  assigned  any  value  at  will,  is 
called  an  independent  variable.  Illustrations  of  an  independ- 
ent variable  are  the  radius  of  a  circle,  the  area  of  a  square, 
in  short,  any  variable  quantity  which  is  quite  arbitrary. 

If  we  determine  that  an  independent  variable  shall  assume 
all  possible  values  which  differ  less  and  less  from  a  constant 
quantity  7,  and  that  this  difference  shall  become  small  at 
will,  we  say  that  we  "let  the  variable  approach  the  limit 
Z."  If  X  denote  the  independent  variable,  this  process  of 
approaching  the  limit  is   indicated  by  x  =  L     The  sign  = 

means    to    "  approacli    the    limit,"    and    x  =  1    is    read,    "  x 

77 


78  CALCULUS  [Ch.  II. 

approaches  the  limit  Z."  As  the  variable  is  entirely  inde- 
pendent, we  can  let  it  approach  any  limit  we  please,  or 
which  the  conditions  of  any  particular  problem  may  lead  us 
to  select.  No  question  can  then  ever  arise  as  to  what  limit 
an  independent  variable  approaches ;  for  it  may  approach 
any  limit  we  choose  to  set.  When  an  independent  variable 
is  defined,  there  is  nothing  more  to  be  said  about  it,  and  it 
accordingly  offers  little  to  interest  us. 

The  case  is  changed,  however,  when  we  consider  some 
second  variable  quantity  whose  value  depends  upon  that  of 
the  independent  variable,  and  is  determined  by  it.  Such  a 
variable  is  called  a  dependent  variable.  The  values  of  a  de- 
pendent variable  are  not  at  all  arbitrary,  but  are  fixed  as 
soon  as  the  values  of  the  independent  variable  are  fixed. 
Given  the  radius  of  a  circle  as  independent  variable,  the 
area  of  the  circle  is  a  dependent  variable ;  likewise  the  cir- 
cumference and  the  diameter  are  dependent  variables ;  given 
the  area  of  a  square,  the  side  and  the  diagonal  are  dependent 
variables.  The  question  at  once  arises:  If  we  let  the  inde- 
pendent variable  approach  some  limit  Z,  what  does  the  de- 
pendent variable  do?  Clearly  the  dependent  variable  may 
change  values  in  consequence  of  changes  in  the  value  of  the 
independent  variable.  These  changes  of  value  may  or  may 
not  be  such  that,  as  the  values  of  the  independent  variable 
differ  less  and  less  from  its  limit,  the  corresponding  values 
of  the  dependent  variable  differ  less  and  less  from  some 
fixed  quantity,  in  such  a  way  that  the  latter  difference 
may  be  made  as  small  as  we  wish,  by  making  the  former 
difference  small  enough.  If  this  is  the  case,  the  fixed 
quantity  from  which  the  values  of  the  dependent  variable 
differ  less  and  less  is  called  the  limit  of  the  dependent 
variable. 


1-2.]  CONCERNING  LIMITS  79 

Art.  2.  Illustrations  of  limits.  I.  Consider  a  triangle 
of  altitude  unity,  and  variable  base.  The  area  of  the  tri- 
angle is  dependent  upon  the  length  of  the  base.  The  length 
of  the  base  is  the  independent  variable,  the  area  is  the 
dependent  variable.  If,  now,  the  length  of  the  base  be 
made  to  differ  less  and  less  from  2,  the  area  differs  less  and 
less  from  unity,  and  can  be  made  to  differ  from  unity  just 
as  little  as  we  please  by  taking  the  base  little  enough  dif- 
ferent from  2.  Thus  the  area  differs  from  unity  by  less 
than  j^Q^  whenever  the  length  of  the  base  is  less  than  j^-q 
different  from  2. 

II.  In  the  same  triangle,  let  the  variable  base  approach 
the  limit  zero.  If  we  take  the  base  small  enough,  the  area 
can  be  made  small  at  will.  Hence  the  area  approaches  the 
limit  zero  as  the  base  approaches  zero. 

III.  Given  a  circle  of  radius  unity,  and  a  concentric  circle 
of  variable  radius  x'^1.  Now,  considering  the  area  of  the 
circular  ring  between  the  two  circles,  we  have  the  area  equal 
to  7r(l  — a;2).  As  x  becomes  more  and  more  nearly  equal 
to  unity,  the  area  becomes  more  and  more  nearly  equal  to 
zero.  By  taking  x  sufficiently  little  different  from  1,  the 
area  may  be  made  to  differ  as  little  as  we  please  from  zero. 
Hence,  again,  as  x  approaches  the  limit  1,  the  area  approaches 
the  limit  zero. 

IV.  Considering  2  a;  —  1,  and  letting  rr  =  —  4,  we  see  that 
2  a;  —  1  may  be  made  to  differ  little  at  will  from  —  9  by 
taking  x  sufficiently  little  different  from  —4.  Hence  2  a:  — 1 
approaches  the  limit  —  9  as  a:  approaches  the  limit  —  4. 

2 

V.  Consider  the  fraction  — -;  as  x  approaches  3,  the 

x^  -{-  1 

denominator  approaches  10,  and  the  fraction  approaches  ^. 
By  taking  x  sufficiently  little  different  from  3,  the  fraction 


80  CALCULUS  [Ch.  II. 

may  be  made  to  differ  little  at  will  from  ^.     Hence  |  is  the 

limit  of  — as  x  approaches  3. 

x^  -}- 1 

Art.  3.  Definition  of  limit.  As  the  precise  definition  of 
a  limit  when  met  for  the  first  time  is  somewhat  difficult  of 
comprehension  or  application,  we  begin  with  several  defini- 
tions, which,  though  loose,  are  often  given : 

1.  The  limit  of  a  variable  is  a  constant  quantity  from 
which  the  variable  may  he  made  to  differ  as  little  as  we  please. 

Nothing  is  stated  here  as  to  how  the  variable  is  to  be 
"made  to  differ."  This  defect  may  be  remedied  by  stating 
the  definition  as  follows : 

2.  The  limit  of  a  (dependent^  variable  is  a  constant 
quantity  from  which  the  variable  may  he  made  to  differ  as 
little  as  we  please  by  choosing  the  values  of  the  independent 
variable  sufficiently  little  different  from  some  quantity  called 
its  limits  which  is  determined  arbitrarily  or  by  the  conditions 
of  the  particular  problem  under  consideration. 

In  still  other  words  : 

3.  if,  as  the  independent  variable  approaches  near  at  will 
to  its  limits  the  dependent  variable  consequently  approaches  near 
at  will  to  some  fixed  quantity^  the  latter  is  called  the  limit  of 
the  dependent  variable^  or  better^  it  is  called  the  limit  which 
the  dependent  variable  approaches  as  the  independent  variable 
approaches  the  limit  fixed  for  it  in  the  particular  discussion 
in  hand. 

These  different  phrasings  of  the  same  idea  are  given  in 
order  that  the  essential  nature  of  a  limit  may  thereby  be 
made  clearer  to  the  student.  As  working  definition,  let  him 
adopt  that  which  is  most  satisfactory  to  himself ;  either  the 
second  or  the  third  will  probably  be  sufficient  for  all  the 
cases  we  shall  take  up.     Still  the  notion  of  a  limit  is  of  such 


i 


2-3.]  CONCERNING  LIMITS  81 

fundamental  importance,  that  we  give  also  an  exact  defini- 
tion, to  which  the  student  may  have  recourse  in  case  the 
definition  which  he  has  adopted  seems  no  longer  quite  satis- 
factory or  applicable. 

4.  Rigorous  definition  of  a  limit.  Given  an  independent 
variable  x  and  a  variable  y  dependent  upon  x  ;  and  considering 
two  numbers  I  and  k  not  involving  x;  if  for  every  positive  num- 
ber^ €  (no  matter  how  small) ^  a  positive  number  B^  exists  such 
that  y  —  I  is  numerically  less  than  e  for  all  values  of  x 
(x^  Jc)^  such  that  x  —  h  is  numerically  less  than  3^,  then  y  is 
said  to  approach  the  limit  Z,  as  x  approaches  the  limit  k. 

Remarks.  1.  The  symbol  8^  is  used  in  this  definition  to  emphasize  by 
the  notation  that  the  vahie  of  8  is  dependent  upon  that  of  c. 

2.  The  restriction,  x  ^  k,  means  simply  that  the  questions  as  to  the 
existence  and  the  value  of  the  limit  may  be  determined  without  taking 
into  account  the  value  or  the  nature  of  y  when  x  —  k. 

3.  The  restriction  x  ^  k  is  equivalent  to  the  two  following  restric- 
tions:  either  x'>k,  ov  x<,k.  \i  x  =^  k  he,  replaced  by  x^k,  we  have 
the  definition  of  the  limit  which  y  approaches,  as  x  approaches  the  limit 
k  through  values  greater  than  k ;  and  if  x  9^  ^  be  replaced  by  x  <  k,  we 
have  the  definition  of  the  limit  which  y  approaches,  as  x  approaches  k 
through  values  less  than  k.  The  two  limits  thus  defined  are  usually  the 
same,  but  not  necessarily  so.  When  this  is  not  the  case  the  function  is 
said  to  be  discontinuous  for  x  =  k,  and  illustrations  will  be  given  when 
the  subject  of  continuity  is  taken  up  (pp.  160-165). 

4.  It  is  not  necessary  that  the  independent  variable  be  free  from  all 
restrictions.  It  is  sufficient  that  it  be  free  to  assume  such  values  as  are 
requisite  for  the  application  of  the  definition.  Thus,  if  arc  sin  x  be  the 
dependent  variable,  the  independent  variable,  a;,  is  subject  to  the  restric- 
tion that  it  may  not  be  numerically  greater  than  unity.  Likewise,  when 
X  grows  large  without  bound,  it  may  often  do  so  through  the  sequence  of 
positive  integral  values :  for  instance,  the  number  of  sides  of  a  regular 
polygon  inscribed  in  a  circle  may  be  taken  as  independent  variable ;  it 
may  grow  large  without  bound,  but  is  always  a  positive  integer. 

It  often  happens  that  the  dependent  variable  approaches 
a  limit,  as  the  independent  variable  increases  without  bound. 


82  CALCULUS  [Ch.  II. 

To  avoid  complexity,  this  alternative  has  not  been  included 
in  the  above  definitions.  It  is  easily  seen  how  the  defini- 
tion of  a  limit  should  be  modified  to  include  this  case. 
Definition  3,  for  instance,  would  read  : 

3a.  If  CIS  the  independent  variable  increases  without  hounds 
the  dependent  variable  consequently  approaches  near  at  will  to 
some  fixed  quantity^  the  latter  is  called  the  limit  which  the 
dependent  variable  approaches  as  the  independent  variable 
increases  without  bound. 

Art.  4.    Application  of  the  definition ;    further   illustra- 

x^  —  4 
tions.     VI.    Let  y   denote   the   fraction  — ,  and   let  x 

X 2 

approach  2.     We  may  write 

^  .      ON  ^  —   2 

whence  we  see  readily  that  when 

x  =  l,  1.5,  1.8,  1.9,  1.99,  1.999,  respectively, 
then 

?/=3,  3.5,  3.8,  3.9,  3.99,  3.999,  respectively. 

As  the  values  of  x  approach  2,  those  of  y  approach  4,  and 
y  may  be  made  to  differ  little  at  will  from  4,  by  taking  x  near 

enough  to  2.     Accordingly  y  or  —  approaches  the  limit 

4  as  a;  approaches  2.* 

*  Recurring  to  the  strict  definition,  we  have  here  I  =  4,  k  =  2  ;   if  e  be 
-,  then  ?/  —  4  will  be  numerically  less  than  provided  x  —  2 


1,000,000  1,000,000 

is  numerically  less  than  ;  i.e.  de  is :  similarly  if  e  be  still 

1,000,000  1,000,000 

smaller,  a  value  5e  exists  such  that  y  —  4  is  numerically  less  than  e,  whenever 

X  —  2  is  numerically  less  than  d^. 


3-4.]  CONCERNING  LIMITS  88 

In   illustration  VI,   if   we   let  the   independent   variable 

actually  reach  the  limit  2,  ^  assumes  the  form  -,  which  may 

have  any  value  whatever.  In  all  the  other  illustrations, 
the  value  of  the  dependent  variable  remains  quite  clear  and 
unambiguous  if  the  independent  variable  is  made  equal  to 
the  limit  iixed  for  it.  As  the  illustrations  have  shown,  this 
is  not  a  material  distinction  ;  the  limits  are  determined  accord- 
ing to  the  same  definition  and  hy  the  same  process  in  each  case. 

The  application  of  the  definition  in  the  determination  of 
the  limit  in  any  specific  case  does  not  require  the  examina- 
tion of  the  expression  in  hand  to  see  what  would  be  its 
character  if  the  independent  variable  were  put  equal  to  its 
limit.*  This  is  expressly  stated  in  definition  4,  and  is 
understood  with  the  others.  Noting,  therefore,  once  for  all, 
that  in  determining  the  limit  of  a  dependent  variable  the 
independent  variable  is  not  to  be  put  equal  to  its  limit,  as 
fixed  in  the  discussion  in  hand,  it  will  be  permissible  to  per- 
form operations  which  would  not  be  valid  without  this 
proviso ;  in  particular,  to  divide  by  a  quantity  which  would 
be  zero  if  the  independent  variable  were  equal  to  its  limit. 

Illustration  VI  can  now  be  treated  as  follows : 

Introducing  the  notation  ^^^^  to  denote  "  the  limit,  as  x 
approaches  the  limit  a,  of  •••,"  we  have 

lim   x'^-4t^  lim   {x  +  2^{x-  2) 
x=2 ^_2      x=2  ^_2 


*  When  this  is  done,  the  function  may  have  a  single  value,  several  values, 
a  boundless  number  of  values,  or  no  value  (being  meaningless);  having  a 
single  value,  this  value  may  or  may  not  be  equal  to  the  limit.  Instances  in 
which  the  limit  and  the  value  are  distinct  will  be  given  in  discussing  con- 
tinuity, (p.  162). 


84  CALCULUS  [Ch.  II. 

Since,  in  accordance  with  our  definition,  x  is  not  to  be 
given  the  value  2  in  the  determination  of  this  limit,  x  —  2 
will  not  assume  the  value  zero  in  this  discussion,  and  we 
may  divide  numerator  and  denominator  by  it,  with  the 
result  that 

Hm    a:2-4_   lim   r^,o\ 

and  the  latter  limit  is  seen  by  inspection  to  be  4. 

The  need  for  the  notion  of  a  Ihnit  is  felt  when  we  have  to  deal  with 
expressions  which  (like  that  in  VI)  lose  definiteness  for  a  certain  value 
of  the  independent  variable.  A  very  common  mode  of  determining  the 
limits  of  such  expressions  is  to  try  to  transform  the  expression  (as  was 
done  in  VI)  so  that  it  is  made  up  of  expressions  which  would  remain 
unambiguous  under  these  circumstances,  and  whose  limits  can  accord- 
ingly be  determined  by  inspection.  This  procedure  will  be  repeatedly 
exemplified  in  subsequent  chapters. 


VII, 


_     lim     X  —  S  _  _  5 

VITI.    To  find  the  limit  of  -  as  a:  grows  beyond  all  limits, 

X  ^ 

we  notice  that  by  taking  x  sufficiently  large,  -  may  be  made 

X 

to  differ  little  at  will  from  zero.     Accordingly,  -  approaches 
the  limit  zero  as  x  increases  without  bound. 

IX.    Similarly,  the  limit  as  x  grows  beyond  all  bounds,  of 

3 
—— -,  is  zero,  since  by  taking  x  sufficiently  large, 

O  2/     ~\~    i  X  —  D 

the  value  of  the  fraction  may  be  made  small  at  will. 


lim 

< 

X' 

'■-x-Q 

X  =  — 

'■^x^- 

■f  7  a;  +  lo' 

lim 

ix 

-3)(a.  +  2)_ 

4-6.]  CONCERNING  LIMITS  85 

X.    To  find  the  limit  of  -— as  x  grows  without  bound, 

0  X  -\-  6 

we  notice  that  for  all  values  of  x  (except  a;  =  0), 

2x  +  l_'^^lc 
bx+'6~  ^      3' 

X 

1  3 

and,  in  the  right  member,  both  -  and  -  approach  the  limit 

XX  fj 

zero  as  x  increases  without  bound,  and  hence  -  is  the  limit 
sought. 

Art.  5.  Concerning  infinity.  When  a  variable,  x^  has 
the  property  of  assuming  values  whicli  grow  larger  and 
larger  without  bound  (Lat.  in-finitus)^  we  often  say,  for 
brevity,  that  "a;  becomes  infinite,"  or  that  "a;  approaches 
infinity"  (symbol,  oo).  These  expressions  mean  neither 
more  nor  less  than  ":r  grows  large  without  bound,"  and  this 
meaning  is  frequently  .denoted  by  the  symbol,  x  =  ao,  which 
may  be  read  in  any  of  the  above  three  forms  indifferently. 
The  terms  infinite  and  infinity  are  always  used  as  abbrevia- 
tions, and  the  full  meaning  of  the  abbreviation  must  be 
clearly  understood.  Infinity  is  not  a  quantity  nor  a  value, 
though  it  is  sometimes  used  with  the  same  phraseology  as 
if  it  were  a  value.  For  instance,  it  is  customary  to  say, 
tan  90°  =  Qo,  log  0  =  —  oo,  etc.  But  though  the  use  of  such 
expressions  may  add  to  compactness  of  form,  it  must  never 
be  forgotten  that  we  are  stating  a  property,  not  a  value,  of 
the  variable  in  question.  This  property  is  that,  under  cer- 
tain circumstances,  the  variable  may  grow  large  without 
bound  ;  the  circumstances  usually  involve  some  considera- 
tions of  limits. 


86  CALCULUS  ICh.  II. 

EXAMPLES 

1.  log  0  =  —  GO 

is  simply  an  abbreviation  for  the  statement  that  when  x  approaches  the 
limit  zero,  logx  is  negative  and  grows  large  numerically  without  bound. 

2.  tan  90°  =  co 

is  an  abbreviation  for :  "  The  tangent  of  an  angle  grows  large  without 
bound  as  the  angle  approaches  the  limit  90°." 

3.  Parallel  straight  lines  meet  at  infinity,  is  merely  an  abbreviation  for 
the  following :  "  Given  a  fixed  straight  line,  and  a  movable  straight  line 
intersecting  it;  if  a  point  P  of  the  movable  straight  line  be  kept  fixed, 
and  the  straight  line  be  turned  about  this  point,  then  the  straight  line 
through  P,  parallel  to  the  fixed  straight  line,  is  the  limiting  position 
which  the  movable  straight  line  approaches,  as  its  point  of  intersection 
with  the  fixed  straight  line  is  moved  to  a  distance  growing  greater 
without  bound." 

Akt.  6.  Further  examples  of  limits. 

YT  li»i>     tan  X  _     lim     sin  x       _  n 

X  _  yu    ggg  ^       X  -vxj    ^Qg  0^ 

lim        . 
=  X  =  90°  ^^^  ^ 

=  1. 

1  +  5-^ 
■     VTT  lim      x^-\-^x-b    ^    lim  X      x^ 

^  =  ^2x^-5x-hl      ^-^2--  +  - 

X      x^ 

1 

=  2' 
since  each  of  the  fractions  in  the  numerator  and  the  denomi- 
nator approaches  zero  when  x  increases  without  bound. 

9 

-^jjj  lim    2:g3_4^2_^9^^    j.^    ^^^ ^ 

^=/»    5x^—6x-^2    ~x  =  cc  0       2* 

X     x^ 


5-7.]  CONCEBNIKG  LIMITS  87 

As  X  grows  large  without  bound,  the  denominator  ap- 
proaches 5,  while  the  numerator  grows  large  without  bound, 
and  hence  the  whole  fraction  grows  large  without  bound. 
In  our  abbreviated  form  we  may  state  this  as  follows  : 

lim    2a^-4:x^-{-9x 


^  =  00    5^2_g^^2 


oo.- 


XIV.  Sometimes  it  is  advantageous  to  pass  to  logarithms 
as  the  first  step  in  the  determination  of  the  limit.  Thus,  to 
find  J^  V3,  we  put 


n  =  co 


y=^, 


and  have  log  «/  =  -  log  3,* 


n=oo      o^        n  =  CO      ^ 


n 
=  0. 

Hence,  Ji"^3^=l.t 

'  n  —  GO  «^  I 

Art.  7.  The  fundamental  theorem  of  limits.  The  idea 
of  limits  is  made  useful  and  available  in  mathematical 
investigations  by  the  following  fundamental  theorem  : 

If  two  variable  quantities  are  always  equal  and  each  ap- 
proaches a  limit,  those  limits  are  also  equal.  $ 

This  theorem  is  almost  self-evident  when  we  understand 
clearly  the  meaning  of  the  expressions  employed  in  it.  If 
two  quantities   are  always  equal,  they  are  identical;  how- 

*  Formula  9,  Appendix.  t  Formula  8,  Appendix. 

\  Of  course,  we  can  speak  of  equality  between  two  expressions  only  when 
both  have  an  unambiguous  meaning.  A  fuller  and  equivalent  wording  of  the 
above  theorem  would  be  :  If  two  variable  quantities  are  always  equal  when- 
ever each  has  a  definite  meaning,  and  if,  lohile  varying  simultaneously^  each 
approaches  a  limit,  those  limits  are  also  equal. 


88  CALCULUS  [Ch.  II. 

ever  the  expressions  may  vary,  they  have  always  the  same 
value  ;   they  can  be  only  different  forms  of  expression  for 
the  same  thing.    Whatever  can  be  said  about  one  can  be  said 
(with  the  proper  change  in  form  merely)  about  the  other. 
For  instance,  \i  x  =  z^^  the  equation 

a:2  -  16  =  2;4  -  16 

is  an  identity  ;  it  is  true  for  all  values  of  z  and  x^  the  latter 
being  fixed  (by  the  relation  x  =  z^^  as  soon  as  z  is  fixed. 
If  we  discover  that  a;  —  4  is  a  factor  of  the  left  member,  it 
follows  without  furtlier  investigation  that  2^  —  4,  expressed 
in  terms  of  z  (i.e.  z^  —  4^,  m  3,  factor  of  the  right  member. 

In  particular,  if  the  variable  expressed  in  one  form  ap- 
proaches a  limit,  the  same  variable,  expressed  in  another 
form,  will  approach  the  same  limit  expressed  in  a  corre- 
sponding form.     This,  then,  is  what  the  theorem  means  : 

If  we  have  two  expressions  for  the  same  variable  quantity^ 
and  if  simultaneously  under  certain  circumstances  each  of  these 
expressions  approaches  a  limit.,  these  limits  can  be  simply  two 
different  expressions  for  the  same  thing. 

As  an  illustration,  let  us  take  the  identity  already  used, 

x^-U  =  z^-U;     (x  =  z^). 

If  X  approaches  the  limit  4,  we  see  that  the  left  member 
approaches  zero.  We  know  then,  from  this  fact  alone,  that 
the  right  member  (being  only  another  form  of  expression  for 
the  left  member),  approaches  zero  also  when  x  approaches  4, 
or,  in  terms  of  2,  when  z^  approaches  4,  or  when  z  approaches 
2  (or  —  2).  Here  we  find  the  limit  of  the  right  member  by 
expressing  the  limit  of  the  left  member  in  the  notation  of 
the  right  member.  But  we  might  just  as  well  have  deter- 
mined independently  the  limit  which  the  right  member 
approaches  when  z  approaches  ±  2  (which  is  equivalent  to 


7.]  CONCERNING  LIMITS  89 

X  approaching  4).  We  know  in  advance  that  the  results 
must  be  equal,  being  the  limits,  under  the  same  circum- 
stances, of  different  expressions  for  the  same  thing.  The 
equality  between  these  limits  may  be  a  relation  of  interest. 
For  instance,  suppose  that  we  had  proved  somehow  that 

x^  —  a  =  z^  —  h 

for  all  values  of  x  and  of  z  subject  to  the  condition  x  =  z"^  \ 
then,  as  x  approaches  zero,  the  left  member  approaches  —  «, 
and  at  the  same  time  the  right  member  approaches  —  h.  We 
have,  therefore,  —  a  =  —  5,  or  a  =  5.  This  is  a  new  relation 
between  the  quantities  a  and  h  which  may  be  of  value. 

As  another  illustration,   consider  the  area  of   a  regular 

AP 

polygon  inscribed  in  a  circle  ;  it  is  ,  where  A  denotes 

the  apothegm,  and  P  the  perimeter  of  the  polygon.     Calling 

AP 

the  area  of  the  polygon  aS',  we  have  S  and  —^—  as  two  differ- 

A 

ent  expressions  for  the  area  of  the  polygon.  But  as  the 
number  of  sides  is  increased,  aS'  approaches  the  circle  as  its 

limit,    and    — —-    approaches    the    limit    !ll— ^,    or    itt^    (r 

denoting  the  radius  of  the  circle).  These  are  two  different 
expressions  for  the  same  thing  (the  limit  of  the  area  of  the 
polygons);  therefore 

Area  of  circle  =  irr^. 

This  fundamental  theorem  adds  one  to  the  ways  in  which 
we  can  deduce  a  new  equation  from  one  already  known. 
We  are  able  to  deduce  new  equations  from  given  equations 
by  various  methods,  such  as  adding  the  same  quantity  to 
both  members  ;  multiplying  both  by  the  same  factor ;  rais- 
ing both  to  the  same  power ;   and  the  like.     In  all  these 


90  CALCULUS  [Oh.  it. 

cases  we  know  that  the  resulting  equation  will  hold  true 
whenever  the  original  equation  does  so.  We  can  now  add 
to  these  another  method  for  deducing  a  new  equation,  m^. 
by  equating  the  limits  of  the  two  members  of  the  original 
equation.  This  method  is  subject  to  the  important  proviso, 
that  the  equation  from  which  we  start  must  hold  true  for 
all  values  of  the  variable  quantity  or  quantities  involved,  — 
must  be  an  identity.  The  other  methods  mentioned  did  not 
labor  under  this  restriction. 

An  identical  equation  having  been  established,  a  new  identi- 
cal equation  can  be  deduced  from  it  by  equating  the  limits  of 
both  members. 

We  say  this  new  equation  is  deduced  by  "taking  the 
limit  of  the  given  equation,"  or  by  "passing  to  the  limit." 
The  resulting  equation  is  just  as  accurate  and  as  rigorously 
deduced  as  that  found,  for  instance,  by  squaring  both  mem- 
bers. This  is  true  because :  The  limit  of  a  variable  (when- 
ever any  exists)  is  a  precise  quantity  and  independent  of  the 
variable.  It  is  not  an  approximation,  but  the  exact  quantity 
to  which,  under  certain  circumstances,  the  variable  approxi- 
mates.* 

This  method  of  deducing  new  equations  is  fundamental  to 
the  applications  of  our  subject. 

Art.  8.  Propositions  concerning  limits.  There  are  cer- 
tain propositions  concerning  limits,  one  or  more  of  which 
must  be  implied  in  almost  every  case  of  the  determination 
of   a  limit.      They  are   quite   plausible  to  beginners,   who 

*  111  some  cases,  variables  may  actually  become  equal  to  their  limits,  in 
others  not ;  but  in  all  cases,  the  variable  may  approximate  closely  at  will 
to  the  limit.  We  shall  see  later  that  this  property  may  be  utilized  to  deter- 
mine, with  any  desired  degree  of  approximation,  the  numerical  value  of 
quantities  proved  (or  defined)  to  be  the  limits  of  certain  variables. 


7-9.]  CONCERNING   LIMITS  91 

usually  tacitly  assume  their  truth,  and  apply  them  without 
having  ever  consciously  formulated  them. 
They  are  the  following  : 

I.  The  limit  of  the  sum  of  a  fixed  number  of  terms  is  the 
sum  of  the  limits  of  the  terms  considered  separately. 

II.  The  limit  of  the  product  of  two  factors  is  the  -product 
of  the  limits  of  the  factors  considered  separately.* 

III.  The  limit  of  a  fraction  is  the  limit  of  the  numerator 
divided  by  the  limit  of  the  denominator,  f 

Proofs  of  these  propositions  will  be  given  in  Art.  11,  p.  92. 

Art.  9.  Concerning  epsilons.  Quantities  which  can  be 
made  small  at  will,  and  which  are  such  functions  of  the 
independent  variable  that  they  do,  in  fact,  approach  zero 
when  the  independent  variable  approaches  the  limit  which 
may  be  selected  for  it  in  the  problem  under  consideration, 
are  often  denoted  by  the  Greek  letter  e,  which  is  read 
"epsilon."  An  epsilon  is  a  quantity  which  approaches  zero 
under  the  conditions  of  the  discussion  in  which  it  occurs.  If 
various  epsilons  occur  in  the  same  discussion,  they  may  be 
distinguished  by  subscripts,  as  e^,  e^,  Cg,  e^,  •••,  e^. 

We  can  express  the  statement  that  I  is  the  limit  oi  x%  (in 
the  form  of  an  equation)  by  use  of  an  e,  viz. 

X  —  1=  €. 


*  There  is  one  exception,  viz.  the  case  in  which  one  factor  approaches 
zero  while,  at  the  same  time,  the  othei-  grows  boundlessly  large. 

t  The  exceptional  cases,  in  which  the  limit  of  the  denominator  is  zero, 
will  be  considered  in  connection  with  the  proof  of  the  proposition. 

t  Such  statements  as  "  Z  is  the  limit  of  cc,"  which  we  shall  often  employ 
for  brevity,  mean  that  the  variable  x  approaches  the  limit  I  when  the  inde- 
pendent variable  approaches  a  certain  limit,  fixed  for  it  by  the  conditions  of 
each  particular  problem. 


92  CALCULUS  [Ch.  II. 

Conversely,  y  —  k  =  e^^ 

expressed  in  words,  is  nothing  other  than  the  statement  that 
the  limit  of  y  is  k.  The  two  forms  of  statement  are  quite 
equivalent. 

Art.  10.  Properties  of  epsilons.  1.  The  sum  of  a  fixed 
number  of  epsilons  is  an  ejjsilon.  To  show  this  we  have  to 
show  that  this  sum  can  be  made  small  at  will.  Let  the  num- 
ber of  epsilons  be  n.  Then,  however  small  the  sum  may 
be  desired  to  be,  it  can  be  made  so  by  taking  each  of  the 

constituent  epsilons  smaller  than  -tli  of  the  desired  sum. 

n 

That  is,  the  sum  in  question  can  be  made  small  at  will ;  it 
is,  therefore,  by  definition,  an  epsilon.  This  result  may  be 
expressed  in  an  equation  as 

^1  +  ^2  +  ^3-^ f-^«  =  €- 

2.  The  product  of  a  constant^  c,  and  an  epsilon^  e,  is  an 
epsilon.  For  however  small  the  product  is  to  be  made,  it 
can  be  made  so,  by  taking  €  smaller  tlian  -  th  of  the  desired 

value.  The  product  can  be  made  small  at  will,  and  is  hence 
an  epsilon. 

3.  The  product  of  any  number  of  epsilons  is  likewise  an 
epsilon.  For  when  each  factor  can  be  made  small  at  will, 
the  whole  product  can  be  made  small  at  will. 

Art.  11.  Proof  of  the  propositions  concerning  limits.    I.  The 

limit  of  the  sum  of  a  fixed  number  of  teimis  is  the  sum  of  the 
limits  of  the  terms  considered  separately. 

Let  x-^^  x^^  Xq,  •••,  Xn  be  the  terms,  and  Zj,  l^,  •••,  l^  the  limits 
which  they  respectively  (and  simultaneously)  approach  under 
the  conditions  of  the  problem. 

Then  we  have  (cf.  p.  91) 


9-11.]  CONCERNING  LIMITS  93 

K^J  ^3  ~"  ^3  ~  ^3' 

We  wish  to  show  that 

(x^  +  2-2  +  iTg  +  •••  +  :?^«)-  (^1  +  ?2  +  ^3  +  -  +  O 
is  an  epsilori. 

Adding  the  equations  (1),  we  have 

(x^  +  ^^2  +  2^3  +  •••  +  a;„)  -  (?i  +  ^2  +  ^3  H ^-  O   • 

=  ^1  +  ^2  +  ^3-1 ^-^«» 

By  the  first  of  the  properties  of  epsilons  proved  above,  the 
right  member  is  an  epsilon ;   which  was  to  be  shown. 

II.  The  limit  of  the  product  of  two  factors  is  the  product  of 
the  limits  of  the  factors  considered  separately. 

Let  the  two  variables  be  x  and  ?/,.  and  I  and  m  their  re- 
spective limits.  Then  we  wish  to  show  that  Im  is  the  limit 
of  xt/.     The  hypothesis  is 

x-l  =  €^,     y  -m  =  e^, 

and  we  wish  to  show  that 

xy  —  lm  =  e. 

We  have  x  =  l  +  e^,     y  =  m  +  e^, 

and  hence  xy  =  Im  -\-  le^  +  me^  +  e^e^-, 

or  xy  —  Im  =  le^  +  ^€2  +  e^Cg. 


94  '  CALCULUS  [Ch.  II. 

The  terms  of  the  right  member  can  each  be  made  small  at 
will,  hence  their  sum  can  be  made  small  at  will,  and  the 
right  member  is  an  epsilon  ;  accordingly, 

xy  —  lm  =  e. 

III.    The  limit  of  a  fraction  is  the  limit  of  the  numerator 
divided  hy  the  limit  of  the  denominator. 

Let  X  and  y  approach  simultaneously  the  limits  I  and  m 

X  I 

respectively.     Then  we  wish  to  show  that  -  approaches  — . 

y  m 

We  have  x—  I  =  e^  and  y  —  m  =  e^^  and  we  wish  to  show 

that 

X  _  I  _ 
y     m 

X  _    I    _    Z  +  €i  I 

y     m      m  -\-  €2     m 

_m(l  +  e^^—  l(m  -f-  e^) 
m  (m  +  €2) 


m(m  -\-  €2) 

In  the  last  fraction  the  numerator  can  be  made  small  at 
will  while  the  denominator  approaches  m^.  If  m  is  not  zero, 
the  fraction  can  therefore  be  made  small  at  will ;  accordingly, 

X  _  I  _ 
y      w 

Exceptional  cases.  (1)  In  case  m  is  zero  and  I  is  not, 
then  in  the  fraction  -,  the  denominator  grows  small  at  will, 

y 

while  the  numerator  does  not;  that  is,  the  fraction  grows 
large  at  will. 


11.]  CONCERNING  LIMITS  95 

(2)  In  case  I  and  m  are  both  zero,  we  cannot  tell  imme- 
diately what  limit  the  fraction  approaches,  but  must  first 
transform  the  fraction  in  some  suitable  manner  before  deter- 
mining the  limit ;  as  was  done,  for  example,  in  illustrations 
VI  and  VII  above. 

EXERCISES    X 

Find  the  hmits  indicated  in  the  following  expressions : 
^      lim    x^  -\-  2x  —  24  ■      /  -         __       lim    J_^ 

•     X  =  4  ^2  _  7  ^  _|.  12'  •    71  =  CO  ^2* 

2      1""     ^x^-^x  ,3  li„j        ^ 

'   x  =  ^  2x^-15  X  ^^• 


«      lim     (x-\-  k)^-x'^  T  /      ,   IX 

3.    ^^^^ ^ ^^       hm  n(n-^l) 


n=^^  (n  +  2)(/2  +  8) 


^      lim    (x^  -  9  y.)3  -  a:6 

*•    r  =  0         ~"^7  16,      lim      3  ar^  -  5 


a:  =  00  0  ^2  _  6  ^ 
x  =  0  o  ^.14  _  3  ^11  +  5  ^5"  ^^       lim    (n  +  l)(n  +  2)(n  +  3) 


g      lim        :ri2  -  3  a:ii  +  x"^ 


g      lim    x^-2x^  -^3x 
'   x  =  0       4:X^-Qx 

n      lim    /^/r^  —  2  arx  +  ar^ 
X  =  r  /,^.^  _  2  hrx  +  6r2' 

o      lini    \  —  x^ 

^=1 1 -X 

Q      Hm    (.r  +  /Q^  -  x^ 

,Q     lim    x^  —  c^ 
X  =  c   X  —  c 

lim    x^  —  \ 


n  =  cc  5  ^3 


11.       .  1 

a:  =  1 


T  o      hm    x:^  —  q^         ,  „   , 


18. 

lim     /1\", 

W  =  GO   \2J   ' 

19. 

lim     /2\« 

20. 

lim                        n* 

n  =  oo  (n  +  l)(2n-l)(l-3n) 

21. 

lim    x^  -  .5  a:5  +  2  a:2 

?? 

lim      a:2  +  4  2-  +  3 

a;  =  -l;^2_7^_8 

23. 

lim  a  —  Vrt2  _  ^r^        j    „     1  . 

8 


Hint.     Rationalize  the  numerator. 


96  CALCULUS  [Ch.  II. 

24.  .   -, Ans.  15. 

x  =  l        {1-xy 

Hint.  Make  use  of  the  theorem  that  if  a  polynomial  vanishes  when 
a  is  substituted  for  x,  then  x  —  a  is  a  factor  of  the  polynomial.  The 
other  factor  may  be  found  by  actual  division. 

25.  ^  n  — == •  Ans.   —  8. 

^  --  y/x-\-2-V'dx-2 

Hint.  Put  x  =  y  -\-  2,  and  find  the  limit  of  the  result  when  y  =  0. 
After  substitution,  rationalize  the  denominator. 

«c         lim      x'^  -\-^x*  -  bx^  -7  x^  -\-^x  +  ii  .        , 

Hint.     Either  the  method  indicated  for  24  or  that  for  25  may  be  used. 


CHAPTER   III 

THE  FUNDAMENTAL  CONCEPTIONS  OF  THE  DIFFERENTIAL 

CALCULUS 

Art.  1.  The  underlying  principles.  The  Calculus  has 
for  its  subject  of  study,  continuous  quantity,  i.e.  quantity 
which  varies  without  a  break  from  one  value  to  another. 
Time  and  motion  are  illustrations  of  continuous  variation. 
Indeed,  the  phenomena  of  nature  are  generally  of  this  char- 
actero  When  a  planet  under  the  influence  of  a  perpetually 
varying  force  revolves  around  the  sun  ;  when  the  air,  in 
propagating  sound,  occasions  by  its  vibrations  ever-changing 
states  of  rarefaction  and  condensation ;  when  by  the  explo- 
sion of  a  mixture  of  hydrogen  and  of  oxygen  gas  the  tem- 
perature rises  very  rapidly  to  a  maximum  only  to  fall 
nearly  as  rapidly,  we  are  always  dealing  with  phenomena 
that  are  varying  continuouslyo  Consequently,  the  careful 
study  of  any  aspect  of  nature  soon  requires  the  application 
of  the  Calculus.     When  Leibnitz  *   and  Newton  f  laid  the 

*  Gottfried  Wilhelm  Leibnitz  (1646-1716)  was  a  man  of  many-sided  genius 
who  left  a  permanent  impress  upon  Philosophy,  Theology,  Philology,  Geology, 
and  other  subjects,  as  well  as  upon  Mathematics.  His  presentation  of  the 
Calculus  appeared  in  a  paper  entitled:  ^'- Nova  methodus  pro  maximis  et 
minimis^  itemque  tangentibus^  quae  nee  fractas,  nee  irrationales  quantitates 
moratur,  et  singulare  pro  illis  calculi  genus^'''  published  in  the  Acta  Erudi- 
torum,  Leipzig,  1684. 

t  Sir  Isaac  Newton  (1642-1727)  made  his  principal  publications  on  our 
subject,  in  the  two  following  works  :  Philosophic  naturalis  principia  mathe- 
matical published  in  1687,  and  Methodus  fluxionum  et  serierum  injinitarum, 
cum  ejusdem  applicatione  ad  curvarum  geometriam,  first  published  in  1736 
(in  an  English  translation),  but  said  to  have  been  finished  in  1671. 

97 


98  CALCULUS  [Ch.  III. 

foundations  of  the  Differential  Calculus,  in  all  probability 
independently,  they  did  not  perhaps  fully  realize  that  an 
aid  to  the  investigation  of  the  problems,  whether  of  pure 
mathematics  or  of  nature,  second  to  none  in  power  and 
fertility,  would,  be  evolved  from  their  ideas.  But  ni  the  two 
centuries  that  have  since  elapsed,  these  ideas  have  not  only 
given  rise  to  a  large  system  of  results  of  the  greatest 
importance  in  mathematics,  but  they  have  also  been  applied 
more  and  more  in  the  various  branches  of  science,  and 
have  extended  over  the  entire  realm  of  physical  phenom- 
ena in  so  far  as  we  have  been  able  to  subject  them  to 
measurement. 

To  develop  an  outline  of  these  far-reaching  methods, 
and  to  show  also  how  the  problems  of  mathematics  and 
the  phenomena  of  nature  may  be  treated  by  their  means, 
is  the  chief  object  of  this  book.  These  methods  are  charac- 
terized by  certain  unique  ideas  and  notions  of  fundamental 
importance.  There  seems  to  be  a  widespread  opinion  that 
they  are  very  difficult  to  understand  ;  but  we  take  occasion 
to  remark  with  emphasis  that  this  is  not  the  case.  With 
precise  formulations,  the  difficulties  vanish  almost  entirely ; 
wherever  they  may  still  occur,  they  are  due  not  so  much  to 
the  notions  and  methods  of  our  subject  itfeelf,  as  to  the 
nature  of  the  problems  or  the  phenomena  to  which  they  are 
applied.  The  mathematical  portion  of  the  discussion  re- 
quires nothing  more  than  the  same  careful  formulation  of 
data  and  hypotheses,  the  same  precautions  in  drawing  con- 
clusions, as  other  branches  of  mathematics,  and  its  results 
are  equally  accurate. 

We  begin  by  discussing  several  problems  whose  solution 
requires  the  application  of  the  underlying  principles  of  the 
Differential  Calculus. 


1-2.] 


THE  FUNDAMENTAL   CONCEPTIONS 


99 


Art.  2.  Motion  on  the  parabola.  Given  that  a  point  moves 
on  a  parabola;  to  calculate  the  direction  of  its  motion  at  any 
instant. 

The  direction  of  motion  changes  at  every  moment,  but 
it  can  be  represented  at  any  position  in  its  path  by  the  direc- 
tion of  the  tangent  ta  the  parabola  at  that  point.  If  we  had 
tlie  figure  of  the  parabola  before  us,  we  could  determine  the 
direction  of  its  tangent  at  any  point  by  actual  measurement ; 
but  the  problem  which  we  have  to  solve  requires  us  to  obtain 
for  the  direction  a  formula  which  is  true  for  all  points. 

For  this  purpose  we  consider  the  parabola  (Fig.  35)  to 
have  the  i/-axis  as  the  axis  of  symmetry.  Its  equation  (in- 
terchanging X  and  y  in  the 


equation  deduced  on  p. 
will  assume  the  form 


21) 


(1) 


x^  =  2  py  or  y 


2p 


O 
Fig.  35. 


Qi 


Let  the  point  P,  for  which      

the   position   of   the  tangent 
is  to  be  calculated,  have  the 

coordinates  x,  y.  Let  the  tangent  at  P  be  ^  and  the  angle 
which  it  makes  with  the  axis  of  x  be  t.  We  have  now  to 
determine  this  angle.  We  can  easily  reach  an  approximate 
result  by  substituting  for  the  parabola  an  inscribed  polygon 
with  a  very  large  number  of  sides,  and  determining  the 
direction  of  the  side  PPj  passing  through  P.  If  P^  has  the 
coordinates  x^y-^^  and  if  a  be  the  angle  which  the  side  PPi 
makes  with  the  axis  of  JT,  it  follows  from  the  right-angled 
triangle  PP^L  that 


(2) 


tan  a 


^P^L^P,Q,-LQ,^y^-y 
PL  QQ^  x^-x 


100  CALCULUS  [Ch.  lit 

Since  P  and*  Pj  are  points  of  the  parabola,  their  coordi- 
nates satisfy  respectively  the  equations 

2/  =  g,andyj=g, 


and  by  subtraction 


_  x^^  —  x^ 


2.p     ' 
when  we  substitute  this  value  in  (2)  we  obtain 

(3)  tan  a  =  —  ^^^  ~  ^^ 

2p   x^  —  X 

If  we  denote  the  distance  QQ^  by  7i,  so  that 

(4)  x^  —  X  =  h  and  x^  =  x  -\-  h, 
then  equation  (3)  becomes 

tan  a  =  —-  ^    7   / 

or,  finally, 

(5)  tan  a  =  -  +  - — 

^      2p 

We  have  thus  determined  the  direction  of  the  side  PPi, 
and  approximately,  the  direction  of  the  tangent.  The  error 
which  we  commit  depends  upon  how  near  the  point  Pj  lies 
to  P ;  that  is  to  say,  it  depends  upon  the  magnitude  of  A. 
It  is  clear  that  we  can  take  the  sides  of  the  polygon  which 
we  have  substituted  for  the  parabola  so  small  that  our  eye 
cannot  detect  a  difference  between  the  figure  of  the  polygon 
and  that  of  the  parabola ;   and  CA^en  so  small  that  our  most 


2.]  THE  FUNDAMENT  AS  ^TZOjy^^,    ,  >     ,     ,  101 

powerful  instruments  of  measurement  cannot  enable  us  to 
detect  any  difference  between  the  tangents  of  the  parabola 
and  the  sides  of  the  polygon. 

When  an  error  is  so  small  that  we  can  neither  see  nor 
measure  it,  it  is  for  all  practical  purposes  not  present ;  from 
the  practical  point  of  view  we  have  therefore  solved  the  pro- 
posed problem.  But  mathematics  demands  perfect  accuracy, 
and  a  simple  consideration  will  now  enable  us  to  make  such 
use  of  the  foregoing  method  that  it  will  give  us  the  abso- 
lutely exact  value  of  the  tangent  of  the  angle.  We  first 
observe,  that  the  right  side  of  equation  (5)  consists  of  two 
terms,  of  which  the  first  does  not  contain  the  quantity  h. 
If  we  now  substitute  for  h  a  series  of  values,  as,  for  example, 
0.1  mm.,  0.2mm.,  etc.,  the  first  term  is  not  changed  at  all; 
only  the  second  term,  which  measures  the  degree  of  approxi- 
mation, is  changed.  If  we  substitute  for  h  smaller  and 
smaller  numbers,  as,  for  example,  0.0000001  mm.,  the  poly- 
gon will  approach  nearer  and  nearer  to  the  parabola.  Of 
course,  we  must  always  distinguish  between  the  polygon 
and  the  parabola,  no  matter  how  small  h  becomes.  We  can 
never,  in  our  thoughts,  bring  the  polygon  to  coincidence 
with  the  parabola,  but  our  mathematical  methods  enable 
us  to  deduce  the  equation  which  must  hold  true  for  the 
parabola  from  that  which  we  know  holds  true  for  the  poly- 
gon ;  all  that  we  have  to  do  is  to  consider  the  limits  which 
both  members  approach  as  h  approaches  zero.  These  values 
are  equal  by  the  fundamental  theorem  of  limits  (p.  87),  so 

that  we  have 

x 

tan  T  =  -• 

P 
We  have,  accordingly,  thus  obtained  the  actual  value  of 
the  tangent  of  the  angle  r,  and  this  equation  represents  the 


102  •     '       ''CALCULUS  [Ch.  III. 

direction  of  the  tangent  to  the  parabola  at  the  point  P,  and 
since  the  point  P  is  ani/  point  of  the  parabola,  it  represents 
the  direction  of  the  tangent  at  everi/  point  with  perfect 
accuracy. 

Art.  3.  Concerning  speed.  The  processes  of  nature  may 
take  place  uniformly  or  with  varying  speed.  We  can  form 
no  clear  conception  of  the  latter,  and  find  it  necessary  to 
express  it  somehow  as  uniform  change. 

If  a  body  moves  uniformly,  we  define  its  speed  as  the  ratio 
of  the  distance  traversed  to  the  time  taken.  But  if  it  moves 
with  varying  speed,  we  assume,  in  order  to  get  an '  idea  of 
its  speed  at  any  given  moment,  that  at  that  moment  it  moves 
uniformly  for  the  brief  interval  of  time  t,  and  in  this  time 
traverses  the  distance  cr;  the  ratio  of  the  distance  a  to  the 
interval  of  time  r  gives  the  mean  speed  with  which  the  body 
moves  over  this  distance.  When  a  body  has  a  varying 
motion,  we  are  accordingly  accustomed  to  define  the  speed  at 
any  moment  to  be  the  speed  which  the  body  would  have,  if, 
at  the  moment  under  consideration,  it  moved  on  uniformly. 
Such  a  procedure  is  strictly  necessary,  for,  as  has  just  been 
said,  our  conception  of  speed  is  limited  to  that  of  uniform 
speed.  When  direction  is  an  essential  element  of  any 
motion,  the  ratio  defined  above  as  speed  is  called  velocity. 

In  general,  we  define  the  speed  of  any  change  in  nature  to 
be  the  ratio  of  the  amount  of  this  change  to  tlie  time  taken, 
with  analogous  definitions  for  mean  speed  and  speed  at  any 
moment. 

Art.  4.  The  motion  of  a  freely  falling  body.  To  determine 
the  velocity  at  any  instant.,  of  a  freely  falling  body. 

When  a  body  falls  vertically  downward  from  a  state  of 
rest,  we  know  that  the  distances  traversed  in  1,  2,  3,  4,  ••• 


W); 


2-4.]  THE  FUNDAMENTAL   CONCEPTIONS  lOB 

seconds  are  C  4|,  9|,  16|,  •••  units  of  length  (^  being  the 

velocity  at  the  end  of  the  first  second  of  fall),  and  that 
in  general  the  distance  I  gone  over  in  t  seconds  may  be 
expressed  by  the  formula 

(1)  i  =  yt\ 

The  velocity  of  the  motion  at  different  instants  is  dif- 
ferent, for  the  distances  traversed  during  any  given  second 

have  lengths  equal  to  ^,  3 '4  5^,  •••,  increasing  continually 

with  the  time.  But,  as  we  have  already  stated  above  (p.  102), 
our  conception  of  velocity  is  limited  to  cases  wherein  equal 
distances  are  passed  over  in  equal  intervals  of  time.  Thus 
we  are  again  confronted  by  the  difficulty  that  the  concep- 
tions which  we  have  to  employ  in  our,  operations  are  not 
directly  applicable  to  the  phenomenon  as  it  actu- 
ally occurs,  and  hence  we  must  have  recourse  to 
a  method  of  approximation. 

To  simplify  matters,  we  substitute  for  the  fall- 
ing body  a  point  having  weight,  for  instance,  the 
weight  of  the  body  concentrated  at  its  center  of 
gravity.  Let  Pq  (Fig.  36)  represent  the  place 
where  the  motion  begins,  and  let  the  falling  point 
reach  the  positions  P,  P^,  P^^  "*•'  ^^^  ^'  ^i'  h  '" 
seconds,  and  let  Z,  Z^,  l^^  -•-  stand  for  the  distances  jPa 

PqP,   Pq^I'    -^^0^2  **'  ti'aversed.      According   to 
(1),  we  have  the  equations 

(2)  I  =  \gt\  I,  =  I  yt^,  \  =  1  ift:^  ....  FIG.  36. 

We  now  imagine  that  there  is  a  second  point  also  moving 
vertically  downward  from  P^,  which  passes  the  positions 
P,  Pj,  P2,  P3,  •••at  the  same  instants  as  the  first  point,  but 


P 
Pi 


104  CALCULUS  [Ch.  hi. 


traverses  the  distances  PP^,  ^1^2'  ^2^Z'  '"  ^^^^^  velocities 
which  are  uniform  throughout  each  distance.  Both  of  the 
points  will  have  different  motions  throughout  these  distances; 
we  shall  see  them  at  any  instant  in  different  positions  within 
these  distances,  although  they  pass  the  positions  P,  Pj,  P^,  ••• 
at  the  same  instants. 

If  8  is  the  length  of  PPj,  and  r  is  the  time  it  takes  for  the 
second  point  to  pass  over  this  distance,  its  velocity  is 

(3)  V=^- 

T 

Now  3  =  PPi  =  PoPi  -  PqP  =h-U 

and,  since  the  positions  P  and  P^  are  passed  at  the  end  of 
t  and  t^  seconds, 

(4)  T=t^-t', 

(5)  therefore,  V=  -  =  ^1^. 

T         t^  —  t 

But  according  to  (2), 
and  by  substituting  this  value  oi\  —  I  in  (5),  we  have 

By  (4)  this  becomes 

or,  reducing, 

(6)  V=gt+lr; 

this  is  the  velocity  of  the  second  point  throughout  the  entire 
distance  PPy 


4-5.]  TBE  FUNDAMENTAL   CONCEPTIONS  105 

We  can  make  the  motion  of  the  second  point  approximate 
as  closely  as  we  wish  to  the  motion  of  the  freely  falling 
point.  The  degree  of  approximation  depends  upon  how 
small  the  distances  PP^,  ^i^v  '"  ^^^  taken. 

But  we  cannot  conceive  of  the  perfect  coincidence  of  the 
motion  of  the  second  point  with  that  of  the  first.  It  is  just 
as  difificult  for  us  to  form  a  conception  of  such  a  coincidence 
as  it  is  to  conceive  of  the  transition  of  a  polygon  into  a 
parabola  ;  still  the  method  of  limits  helps  us  out  again  ;  if  we 
determine  the  limit  which  the  velocity  of  the  auxiliary  point 
approaches  as  r  approaches  zero,  that  limit  is  the  exact  value 
of  the  velocity  v  at  the  time  t.     We  find  thus 

a)  v=gt, 

and,  since  P  is  an  arbitrary  point,  this  formula  defines  the 
velocity  of  a  freely  falling  body  at  every  moment  of  its 
motion. 

Art.  5.  The  linear  expansion  of  a  rod.  To  ascertain  how 
a  rod  expands  at  any  moment  while  being  heated.  Experi- 
ment shows  that  a  rod  whose  length  at  the  temperature  of 
melting  ice  we  may  call  unity,  on  being  heated  expands 
in  such  a  way  that  its  length  I  at  any  temperature  6  can  be 
represented  by  the  expression  a. 

(1)  ;  =  1  +  5l9  +  ce\ 

in  which  h  and  c  stand  for  constant  numbers  that  depend 
upon  the  nature  of  the  rod,  and  can  be  determined  experi- 
mentally. 

We  define  first,  the  coefficient  of  expansion  at  any  tempera- 
ture ^,  as  the  increase  in  length  which  the  rod  would  undergo 
during  a  rise  of  one  degree  in  temperature  above  ^,  if  the 


106  CALCULUS  [Ch.  lit 

expansion  continued  uniformly  in  this  interval.     We  seek  to 
determine  the  coefficient  of  expansion  of  our  rod. 

Let  Z,  ?j,  ^3  •••  be  the  lengths  of  the  rod  corresponding  to 
the  temperatures  ^,  6^^  ^2'"'  ^®  ^^^^  suppose  that  the 
expansion  occurs  uniformly  when  the  rod  is  heated  from  6  to 
^j,  from  6^  to  6^,  etc.  The  rod  accordingly  increases  uni- 
formly in  length  from  I  to  l^  when  it  is  warmed  from  ^  to  ^^ ; 
hence,  the  expansion  A^  which  corresponds  to  a  rise  in  tem- 
perature of  one  degree,  is 

According  to  (1) 

and  by  subtraction 

1^-1  =  h(e^  -  (9)  +  c(e^^  -  ^). 
If  we  put 

(3)  Zi  -  Z  =  X,    6>i  -  6>  =  e, 

so  that  \  is  the  increase  of  length  corresponding  to  a  rise  of 
temperature  equal  to  @,  and  bear  in  mind  that 

we  have 

or  finally,  after  substitution  oi  0  -{- &  for  0^ 

(4)  A  =  b-\-c(2e-^S}  =  d-\-2ce-\-c%. 

This  is  the  coefficient  of  expansion  for  the  difference  of 
temperature  (^^  —  6),  it  being  assumed  that  the  expansion  is 
uniform. 

The  smaller  we  make  S  (the  difference  between  6^  and 


5-6.]  THE  FUNDAMENTAL   CONCEPTIONS  107 

^),  the  nearer  our  representation  of  the  process  of  expansion 
approximates  to  its  actual  course  at  the  moment  when  the 
temperature  is  0.  While  here,  as  in  the  previous  instances, 
a  complete  coincidence  cannot  be  conceived,  the  formula 
gives  the  correct  result.  We  determine  the  limit  which  A 
approaches  as  @  approaches  zero,  and  thus  get  for  the 
required  coefficient  of  expansion  a  at  the  moment  the  tem- 
perature is  equal  to  0 

(5)  a  =  h  +  2ce; 

since  0  may  have  any  value,  this  formula  holds  for  the  entire 
course  of  the  process. 

It  is  worth  while  to  remark  that  the  coefficient  of  expansion  can  also 
be  defined  as  a  speed.  .  It  is  the  measure  of  the  lapidity  with  which  the 
increase  of  length  takes  place  when  the  heating  occurs  uniformly.  We 
have  just  as  clear  an  idea  of  the  rapidity  of  a  process  as  we  have  of  the 
rapidity  of  a  motion.  The  increase  of  length  per  degree  accompanying 
a  uniform  rise  of  temperature  corresponds  perfectly  to  the  increase  in 
distance  traversed  by  a  moving  body  per  second ;  in  accordance  with  this 
the  quantity  a  may  be  termed  the  speed  of  expansion  at  the  tempera- 
ture B.  y  * 


Art.  6.  The  derivative.     The  determination  of  the  tangent 
of  the  parabola  was  based  upon  the  equation 

(1)  tan«  =  ^i^  =  i^  +  ,^         ■ 

x^  —  xpAp 

from  which  by  taking  the  limits  of  both  members  we  obtained 

(2)  tanr^-."^ 

This  corresponded  to  our  making  the  polygon  approach  the 
parabola  by  diminishing  the  lengths  of  its  sides.  There  is 
another  notation  which  is  much  used  and  which  is  sometimes 


108  CALCULUS  [Ch.  III. 

more  convenient.  In  the  problem  of  motion  on  the  parabola 
we  now  denote  the  difference  of  the  abscissae  by  Aa;  instead 
of  by  h  and  the  difference  of  ordinates  by  A?/;  that  is,  we 
pat 

(3)  x^  —  x=  ^x   and    y^—  y  —  A^, 

so  that  x^=^  X  -\-  Ax   and   y^  =  y  -{-  Ay. 

Hence  Ax  and  Ay  signify  the  increments  which  x  and  y 
receive,  respectively,  when  we  pass  from  the  point  P  to  the 
point  Pj,  and 

(4)  tan  «  =  ^1^=4^; 

x^  —  X       Ax 

the  limit  of  this  quotient,  as  Ax  =  0,  is  the  value  of  tan  r  ;  it 
is  called  the  derivative  (and  also  differe^itial  coefficient)^  or, 
more  exactly,  the  derivative  of  y  with  respect  to  a?,  and  is 

represented  by  — .     We  have  therefore  for  the  parabola 
(tx 

(5)  tHnT  =  ^'  =  ?;. 

ax     p 

The  case  of  freely  falling  bodies  is  entirely  analogous. 
We  started  from  the  equation 

(6)  V=^f^  =  gt+(Lr. 
In  this  case  we  j)ut 

(7)  t^-t  =  At  and  1^-1=  Al, 

so  that  A^  indicates  the  increment  of  distance  for  the  incre- 
ment of  time  A^ ;   then  the  limit  of  the  quotient 

(8)  F=-^i— =  ^^1 
^  ^  t^-t     At 

when  A^  =  0,  yields  the  value  of  v. 


6-7.]  THE  FUNDAMENTAL   CONCEPTIONS  109 

As  in  the  previous  case,  we  call  the  limit  which  —  ap- 
proaches, the  derivative ;  or,  more  accurately,  the  derivative 

of  I  with  respect  to  ^,  and  write  for  it,  — ;  so  that  for  freely 
falling  bodies  the  equation  holds 

(9)  .=!=,. 

And,  finally,   in   the   last   example   which   we   discussed 
above,  we  began  with  the  equation 

(10)  A  =  ^^=h  +  2ce-^c(&, 

and  found  that  the  limit  of  the  quotient  as  @  approaches  zero 
was  an  expression  giving  the  coefficient  of  expansion,  viz. 

(11)  '        a=b-\-2ce. 
Here  again  we  put 

(12)  1^-1=  M,  e^-e  =  A(9, 

so  that  AZ  indicates  the  increment  of  length  corresponding 
to  the  increment  of  temperature  A^,  whence 

(13)  A  =  —' 

As  in  the  previous  cases,  we  call  the  limit  of  this  quotient 
when  A^  =  0  the  derivative  of  I  with  respect  to  ^,  and  denote 

it  by  -— ,  so  that, 

(14)  «  =  *  +  2..  =  |. 

Art.  7.  The    physical   signification  of  derivatives.      The 

foregoing   examples   may  serve   to   show  how  various   the 


110  CALCULUS  [Ch.  III. 

problems  are  to  which  derivatives  may  be  applied  ;  we  may 
even  say  that  students  of  science  often  make  use  of  deriva- 
tives unwittingly.  Thus,  as  the  derivative  of  a  distance 
with  respect  to  the  time  in  which  it  is  traversed,  expresses 
the  speed  with  which  the  given  distance  is  traversed,  so  the 
derivative  of  the  amount  of  a  substance  reacting  chemically, 
with  respect  to  the  time,  expresses  the  speed  of  reaction.  If 
we  are  considering  the  relationship  between  the  temperature 
and  the  volume  of  a  liquid,  or  the  length  of  a  rod,  or  the 
electromotive  force  of  a  voltaic  cell,  the  derivatives  of  these 
quantities  with  respect  to  the  temperature  are  their  tempera- 
ture coefficients.  If  a  metal,  as  iron,  be  subjected  to  the 
action  of  a  magnetic  field,  the  metal  itself  becomes  mag- 
netized ;  that  is,  it  acquires  a  certain  magnetic  moment. 
The  derivative  of  this  moment  with  respect  to  the  inten- 
sity of  magnetization  is  called  the  capacity  for  magnetiza- 
tion of  the  metal  in  question,  and  characterizes  its  magnetic 
behavior. 

Art.  8.  The  function-concept.  When  the  pressure  to 
which  a  gas  is  subjected  is  altered,  the  volume  occupied 
by  the  gas  also  changes,  expanding  or  contracting  accord- 
ing as  the  pressure  diminishes  or  increases.  .The  relative 
change  of  pressure  and  volume  takes  place  in  accordance 
with  Boyle's  Law  (p.  3),  and  the  interdependence  between 
pressure  and  volume  comes  under  the  concept,  which  is 
known  in  mathematics  as  the  function-concept,  and  is  defined 
as  follows : 

The  quantity  y  is  a  function  of  the  quantity  x.,  if  x  and  y 
are  so  related  that  to  every  value  tvhich  x  may  assume  there 
correspond  one  or  more  values  of  y. 

Hence  we  speak  of  the  volume  of  a  gas  as  being  q.  function 


7-8.]  THE  FUNDAMENTAL   CONCEPTIONS  111 

of  the  pressure  to  which  it  is  subjected.  Similarly,  we  speak 
of  the  solubility  of  a  substance  as  being  a  function  of  the 
temperature,  and  the  diameter  of  a  soap  bubble  as  being  a 
function  of  the  pressure  of  the  air  within  it.  Likewise,  th.e 
law  of  freely  falling  bodies  expresses  a  relationship  between 
the  distance  traversed  and  the  time  in  which  it  is  traversed, 
and  therefore  the  distance  is  a  function  of  the  time. 
Boyle's  Law  may  be  expressed  by  the  equation 

(1)  vp  =  v^Pq,  or  V  =  -^, 

where  Vq  and  p^  are  the  values  of  the  volume  and  pressure 
of  the  gas  in  its  initial  state. 

Similarly,  for  bodies  falling  from  rest,  we  have  the  equa- 
tion 

(2)  i=yt'^ 

where  I  is  the  distance  traversed  in  the  time  ^,  and  ^  is  a 
constant. 

These  equations  enable  us  to  calculate  for  every  value 
of  p  the  corresponding  value  of  v,  and  for  every  value  of 
t  the  corresponding  value  of  I ;  accordingly  v  is  a  function 
of  p,  and  Z  is  a  function  of  t. 

But  we  have  only  to  put  the  above  equations  into  the 
forms 

(3)  p^Pf  and  t  =  Vp, 

to  recognize  that  the  pressure  p  is  also  a  function  of  the 
volume  V,  and  t  is  a  function  of  I ;  for  from  these  equations 
we  can  calculate  the  values  of  p  and  t  corresponding  to  any 
values  assigned  to  v  and  I.  Which  of  these  two  forms  of 
expression  should   be  selected  depends  upon   the   problem 


112  CALCULUS  [Ch.  III. 

with  which  we  are  dealing,  and  the  form  of  the  result  we 
seek.  In  the  first  case,  p  and  t  were  regarded  as  inde- 
pendent variables,  and  v  and  I  respectively  as  dependent 
variables  (pp.  77-78),  while  in  the  second  case,  we  chose  to 
regard  v  and  I  as  the  independent  variables,  and  conse- 
quently p  and  t  as  variables  dependent  upon  them.  In  the 
processes  of  nature  that  require  time  for  their  completion, 
it  is  customary  to  regard  the  time  t  as  the  independent  vari- 
able, since  we  feel  that  the  time  passes  in  a  constantly 
uniform  way  which  is  entirely  independent  of  ourselves,  and 
may  therefore  well  be  regarded  as  a  "  natural "  independent 
variable.  But  nothing  prevents  us  from  choosing  t  as  the 
dependent  variable  for  the  purposes  of  calculation,  just  as 
was  actually  done  in  equation  (3) ;  we  can,  for  example, 
take  up  the  problem :  to  determine  the  time  t  required  by  a 
falling  point  to  traverse  a  given  distance. 

Another  illustration  may  be  taken  from  Analytic  Geometry.  In  every 
equation,  between  the  coordinates  x  and  y,  which  represents  a  curve,  x 
and  y  are  variable  quantities;  they  can  assume  a  countless  number  of 
sets  of  corresponding  values,  and  their  changes  are  regulated  by  the  law 
expressed  algebraically  by  the  equation  in  question  (and  graphically,  in 
the  curve  corresponding  to  it). 

If  X  be  taken  as  the  independent  variable,  then  y  is  the  dependent 
variable;  by  means  of  the  given  equation,  a  value  of  y  can  be  found 
corresponding  to  every  value  of  x,  and  therefore  y  is  a  function  of  x. 
But,  on  the  other  hand,  with  the  aid  of  the  same  equation  we  can  calcu- 
late for  any  value  of  y  the  corresponding  value  of  x,  that  is  to  say,  we 
can  also  regard  a;  as  a  function  of  2/,'or  consider  y  as  the  independent, 
and  x  the  dependent  variable.  These  sets  of  values  enable  us  to  plot  the 
curve,  which  is  the  same  whether  we  determine  the  ordinate  correspond- 
ing to  each  abscissa,  or  vice  versa. 

We  have  already  become  familiar  in  elementary  mathe- 
matics with  the  simplest  functions,  such  as  powers,  loga- 
rithms, trigonometric  functions: 


8.]  THE  FUNDAMENTAL   CONCEPTIONS  113 

rr",  log  X,  sin  x,  cos  a;,  tan  x^  cot  x,  etc. ; 

by  combining  these  we  can  obtain  a  large  number  of  new 
functions,  as,  for  example, 

-,    Vl  +  a;2,    log     ~    ,   sin  x  -f-  cos  x,  etc. 

X  a  +  x 

« 

As  symbols  for  functions  of  x,  the  signs 

/(a;),  <l>Cx),  F(x\  L(x\ 

and  the  like  are  in  general  use,  and  others  may  be  intro- 
duced as  occasion  demands.     Thus  the  equations 

(4)  y  =/(a;),  s  =  (^(O?  ^  =  L(u)^  etc., 

mean  that  y  is  some  function  of  x^  s  is  some  function  of  t,  w 
is  some  function  of  w,  etc.  If,  then,  x^y^^  ^2^2'  ^^s'  ^tc, 
are  corresponding  values  of  x  and  y,  this  is  expressed  by  the 
equations 

(5)  y^  =/(^i),  3/2  =/(^2)'  Vz  =f(^s)^  etc., 

a  mode  of  expression  with  which  we  have  become  familiar 
in  Analytic  Geometry. 

As  examples  of  functions  taken  from  nature,  we  men- 
tion the  following:  The  tension  of  a  vapor  is  a  function  of 
the  temperature;  the  time  of  vibration  of  a  pendulum  is  a 
function  of  its  length ;  the  strength  of  an  electromagnet 
is  a  function  of  the  strength  of  the  electrical  current  and 
of  the  number  of  windings  of  the  wire ;  the  properties  of  the 
chemical  elements  are  functions  of  their  atomic  masses ; 
the  temperature  at  which  water  boils  is  a  function  of  the 
atmospheric  pressure,  etc.,  etc. 


4U4  ^y\^^^^^^ 


114  4  ( A  -^  ^  r   KrCALCl/LUS  [Ch.  III. 


EXERCISES   XK 

1.   If  (i.)  f(x)  =  x%  form  f(x  +  h).  Ans.  x^  +  2xh  +  h\ 

(ii.)  f{x)  =  sin  a:,  form  f{x  -\-h).  Ans.  sin  (x  +  h). 

(iii.)  f(x)=  \ogx%  form  /(x  +  A).  ^4ns.  log  (a:  +  h^). 

(iv.)  /(x)  =  x^  +  2x-6,  form  /(2).  .bis.  3. 

(v.)  <f>(x)  =  a:2  +  2,  form  <^(«  +  b). 

"^  2.   If  F(7/)  =^2/^  -  2 2^2  4.  7 2^  _  9^  show  that 

^  (ii.)   F(-l)  =  -21.  \/;  j,//>\      3ft3  ,4^2^08  fe- 72 

(iii.)    F(2)=21.  ^    '^      V2;  8 

(iv.)  F(0)=-9. 
'  (vii.)  F(y  +  h)=3y^-2y^+7i/-d+(9f-iy  +  7)h+(9y-2)h'^  +  ^h^ 

3.  If  <^(2)  =  ^2  _  9  2;  +  20,  show  that 

(i.)       «^(l)-f<^(0).  (iv.)  <f>(z  +  2)  =  <f>(z)-<f>(fy)-<t>(l)+iz. 

(ii.)       <^(4)-<^(5).  (V.)  <f>(z  +  k)=<f>(z)  +  (2z-d)k  +  k^. 

(iii.)<^(-2)  =  7<^(2). 

4.  If  /(0  =  ^, 

show  that  1 =  "^-^-^  +  ^. 

/(«)+/(&)  2a6-2 

5.  If  <f>(x)  =  log  ^^-^,  show  that 

V(.:).,(,)=,|i5£^[.  ■ 

6.  If  F(y)  =  ?/2n  +y2r^i^  show  that 

F(a)  =  F(-a). 

7.  If  <^(w)  =  m2h+i  +  u^r+i  +  w3  _  5  ^^  ghow  that 

cl>(u)=-<t>(-u). 

8.  If /(a:)  =  sin  X,  show  that 

/w=-/(-^)- 


8-9.] 


THE  FUNDAMENTAL   CONCEPTIONS 


115 


9.   If  i(/(x)  =  cos  (3  x),  show  that 

i{/(x)=il/(-x). 

10.    Assuming  a  curve  as  the  graph  of  y  =f{x),  what  would  be  the 
graph  of  y  =  —f{x)  ? 

Art.  9.  General  rule  for  the  formation  of  derivatives.     Let 


(1) 


y=f(p^^ 


be  any  function  of  x,  and  let  the  accompanying  geometric 

curve  (Figo  37)  be  its  graph.     We 

take   up   the   problem  to  find  the 

tangent  to  the  curve  at  any  of  its 

points. 

We  use  again  the  method  em- 
ployed, pp.  99-101.  If  we  imagine  a 
polygon  having  its  vertices  P,  P^, 
P^^  "-  lying  on  the  curve,  we  can 
easily  determine  the  angle  a  which 
the  side  PPj  of  the  polygon  makes 
with  the  axis  of  abscissae.     We  get 


Y 

P 

/ 

f 

T 
L 

0 

Q      .( 

h 

Fig.  37. 


tan  a 


PL       OQ^  -  OQ      x^ 


or,  inasmuch  as  y  =  f(x)  and  y-^  =  f(x^^ 


(2) 


tan  a  = 


X-t  X 


By  putting  x^  —  x=  h^  this  quotient  may  be  transformed 
so  that  it  will  contain  only  x  and  A,  assuming  the  form 


(3) 


tan  a 


fix  +  K)-fix-) 


116  CALCULUS  [Ch.  hi. 

The  angle  r,  which  the  tangent  at  P  makes  with  the 
axis  of  X,  is  the  limit  which  the  angle  a  approaches  as  h 
approaches  zero. 

We  cannot  actually  determine  the  value  of  this  limit 
unless  the  function  in  question  is  given,  but  can  merely 
indicate  it  by  the  expression 

...  Hm    r/(^  +  A)-/(^)-] 

W .  h  =  0  [  I  J- 

Therefore  the  direction  r  of  the  tangent  at  every  point 
of  the  curve  represented  by  equation  (1)  is  given  by  the 
equation 

(5)  ^„^..p-^ »)-/(»)]. 

Denoting,  as  on  p.  108,  the  difference  of  abscissae  by  Aa;, 
and  the  difference  of  ordinates  by  Ay ;  that  is,  putting 

(6)  x^-x=-^x,   y^-y  =  ^y, 
we  obtain  ^  />.  . 

^^  x^  —  X      Ax        Ax 

and  this  is  the  quotient  whose  limiting  value  for  ^  =  0,  or 
for  Ax  =  0,  is  to  be  determined.  The  fraction  represents  the 
ratio  between  the  increment  of  the  function  and  the  increment 
of  the  independent  variable;  it  is  accordingly  a  measure  of 
the  greater  or  less  rapidity  with  which  the  function  increases 
or  diminishes.  The  limit  of  this  ratio  is  what,  in  previous 
instances,  we  have  already  called  the  derivative  ;  or,  ex- 
pressed more  exactly,  it  is  the  derivative  of  «/,  or  /(a?), 
with  respect  to  a?,  and 

ttJU  fA/JU  UJU  lA/tMj 

are  symbols  each  of  which  is  often  used  to  denote  this  limit. 


9.]  THE  FUNDAMENTAL   CONCEPTIONS  117 

The  symbol  -^  is  not  a  fraction,  of  which  dt/  is  the  numera- 
tor and  dx  the  denominator,  but  denotes  the   limit   of  the 

fraction  — ^. 

Ax 

d 

The  symbol  —  placed  before  any  function  /(a?)  denotes 

dx 

that   the   following  operation  is   to   be    performed   on   that 

function  i 

First,  the  fraction  •ZA^-JI — ;  ~J\^)   {^  to  be   formed,  and 

h 
then  the  limit  of  this  fraction  as  h  approaches  zero  is  to  be 

taken. 

We  have  accordingly  the  defining  equatioii 

/ex  dfjoc)  _   lim   r/(ar;  +  ^)-/(a3)1 

^^^  ~d^-h  =  (^V  n  J' 

or  we  may  also  write  it,  putting  y  in  place  oif(x)  for  brevity, 

(^\  ^_     lim     % 

^  ^  dx      A2=0Aa;° 

The  result  of  the  foregoing  discussions  concerning  the 
tangent  may  now  be  stated  thus  :  For  every  curve  whose 
equation  is  given  in  the  form 

y  =/(^). 

the  direction  of  the  tangent  at  any  of  its  points  is  determined 

by  the  equation 

dy      df(x) 
tan  T  =  -/  =  •^;     - 
ax        ax 

The  definition  of  the  derivative  should  be  firmly  fixed  in 
mind,  both  as  given  in  (8)  and  also  as  expressed  in  words  : 
The  derivative  of  any  function  with  respect  to  a  variable  is 


118  CALCULUS  [Ch.  m. 

determined  hy  means  of  that  fraction  whose  numerator  is  the 
difference  between  the  value  of  the  function  when  the  variable 
receives  an  increment  and  the  value  of  the  function  as  given^ 
and  whose  denominator  is  the  increment;  the  derivative  is 
the  limit  which  this  fraction  approaches  as  the  increment 
approaches  zero.     Thus  the  derivative  of  a^  is 

Urn    (^x  -\-  hy  —  cc^ 
h  =  0  I  ' 

that  of  sin  x  is 

lira    sin  (x  +  /g)  —  sin  x      . 
k  =  0  I  '  ^^^• 

The  letters  used  are,  of  course,  immaterial. 

This  definition  is  fundamental  for  our  whole  subject ;  it 
gives  us  a  general  scheme  or  rule  according  to  which  to  form 
the  derivative  of  every  function.  One  of  the  first  problems 
that  we  shall  solve  is  to  determine  the  derivatives  of  the 
different  simple  functions.  We  know  the  derivative  onl^ 
in  case  we  can  find  the  limit  indicated.  If  in  any  particular 
case  no  definite  limit  exists^  the  function  in  question  has  no 
derivative. 

The  process  of  finding  the  derivative  of  any  given  func- 
tion is  called  differentiation. 

EXERCISES   XII 

Write  the  defining  expression  for  the  derivatives  of 

x\   1,  (a:-3)(x2  +  5),   ^^,  logy. 
X  X  —  6 

2.   Find  the  derivative  for  the  first  three  of  the  expressions  in  1. 
Of  what  functions,  and  with  respect  to  what  variables,  are  the  fol- 
lowing expressions  derivatives  ?    (Answer  by  inspection.) 

«      lim    <j>{x  +  h)-  <l>(x)  lim    Va:  +  A  -  V^ 

'^'  h  =  o  h  ^'  h  =  o 1 

lim  il/(u-hk)-i};(u)  lim    F(y  +  l)-F(y) 

*•    k=0  k  ~'  1  =  0  I 


9.]  THE  FUNDAMENTAL   CONCEPTIONS  119 


7. 


lim  f(t  +  z)-f(z) 
t  =  0  't 


lini    sin  (x^  -j-  h)  —  sin  x^ 
^'   h  =  0  h 

Q      lim    log(xH2^a:  +  ^^)-loga:2 
^-   ^  =  0  d 

lim    <^  (a^  +  a  +  ?/)  -  <^  (a^  +  a) 


10. 


=  0 


2/ 


T,      lim    <^{(a  +  2/)3  +  a  +  2/}-<^(a3  +  a) 
y  =  0  y 

lim   tan  {x  +  c^)  —  tan  x 

j^3       lim     f{x  -i-mr)-f(x)  ^^       lim      <^(g  +  Am)  -  <^  (s) 

•  mr  =  0  mr  '  *  Am  =  0  Am 

,-      lim    cos(m  + A'^4- A.)-cosM  lim    /(^^  +  p^)  - /(^2) 

•^*-  X  =  0  AH  \  '  ^^'  p  =  0  p2 

17.   Write  the   answers  to  3  •••  16  in  the  notation  for  derivatives 
explained  above. 


CHAPTER   IV 

DERIVATIVES  OF  THE   SIMPLER  FUNCTIONS 

Art.  1.  The  derivative  of  oc^.  The  derivative  of  the 
expression  a;",  n  being  a  positive  integer,  is  found  as  follows  : 
The  ratio 

h 
has  in  the  present  case  the  value 

{x  +  hy  —  x^ 


h 
which  by  the  binomial  theorem  is 

CD 1 ^:!_^ 1__ _. 

This  is  the  fraction  whose  limit  is  the  derivative  ;  x^  and 
—  x^  in  the  numerator  cancel  each  other,  and  if  h  be  taken 
out  of  the  parenthesis, 

^  ^         .  h 

^  h 

120 


1-2.]        DERIVATIVES   OF  THE  SIMPLER  FUNCTIONS       121 

If  A  be  now  made  to  approach  the  limit  zero,  the  right  mem- 
ber approaches  the  value  nx^~^  as  its  limit;  the  derivative  of 
x^  is  therefore  nx"~^.     We  have  thus  the  equation 

(3)  ^(^  =  ^a;»-i. 

doc 

To  illustrate,  the  derivative  of  x^  is  2  x^  that  of  x^  is  3  a;^, 
etc.  In  particular,  it  follows  that  the  derivative  of  x  itself 
is  1,  as  is  directly  evident  also,  since  when  n  =  1 

X  -\-  h  —  x      -, 


Art.  2.  The  derivative  of  sin  oc  and  of  cos  op.  To  obtain 
the  derivative  of  sin  x^  we  first  form  the  fraction  whose 
limit  is  the  derivative,  viz.  : 


o    •    X  ^h  — Nr  X  -\-  h  +  X 

2  sin — '-V '  .  cos — —z-^ — - 

C<ni    ^  ryf    _J_     Jt\   Qm    'T' 


sin  (x  +  h)  —  sin  x  '2  2 


h 

sm 


2       f        h\ 

-cos^a;  +  ^j. 


h 

2 

.    h 
sin- 
But  the  limit  of  —j—,  as  h  approaches  zero,  is  unity,f  and 

2 
that  of  cos  [a;  +-)  is  cos^c,  and,  accordingly,  the  right  mem- 


*  Formula  41,  Appendix. 

t  This  is  usually  proved  in  works  on  trigonometry.     It  may  be  proved 
as  follows,     ki  Fig.  38  it  is  seen  without  difficulty  that 

triangle  BOB'  <  sector  BAB'0<  triangle  TOT'. 


122 


CALCULUS 


tCn.  IV. 


ber  of  equation  (1)  has  the  limit  cos  re,  when  h  approaches 
the  limit  zero ;  hence  the  derivative  of  sin  x  is  cos  x^  or 


C2) 


d  sin  PC 
dx 


=  cos  a?. 


In  Fig.  39  is  shown  the  curve  whose  equation  is 

^  =  sin  X. 


Let  the  angle  AOB  contain  -  radians  (p.  74)  ;  then 


BOB'  =  \BB'  'OC=BC  -00=  OB 


2     BC     OC 
Ob'  OB 


sector  BAB'O  =  \  OB^  arc  BB'  =  r'^  -• 


r^  sm  -  cos 


oh 


TOT'  =  I  TT  '  OA=AT-  0A  =  OA^  •  ^^  =  rHan  -. 
^  OA  2 


Fig.  38. 
The  inequality  above  therefore  becomes 


.    h 

,y  .    h        h        ,^h        „        2 
H  sin  -  cos  H  <  H  -  <  r^ -' 


cos 


or,  after  division  by  r^  sin  - ,  into 

2 


'"''-2<—h< 


sm  -      cos 

2 


*  Formula  61,  Appendix. 


I 

'■^  2.] 


DERIVATIVES   OF  THE  SIMPLER  FUNCTIONS        123 


If  it  be  borne  in  mind  that  (p.  117) 

,             dy      dsh 
tan  T  =  -f-  = 

dx  d: 

it  is  easily  seen  that  to  the  values 


,  dy      d^mx 

tan  T  =  -^  =  — - —  =  cos  a;, 
dx         dx  . 


T- 

- 

Y 

!Y 

/ 

1 

1 

/^ 

r" 

•N 

V 

P 

y 

^ 

^s 

P 

( 

,/ 

•^ 

/ 

1 

> 

V 

V 

/ 

\ 

/ 

/ 

0 

ir 

16' 

A 

g 

TT 

/a^ 

571 

37T 

\ 

""TT 

/ 

kk 

X 

y 

« 

! 

\ 

^ 

3 

^ 

/ 

2 

vj 

^ 

3 

X* 

'U- 

! 

1 

L_ 

1_ 

J 

Fig.  39. 

there  correspond  the  values 

^  =  0,  1,  0,   -1,  0,  1,  0,  ..., 
and 

tan  T  =  cos  0,  cos  — ,  cos  tt,  cos  -— -,  cos  2  tt,  cos  -— ,  cos  3  tt, 


=   1,    0,  -1, 


0, 


1, 


0, 


The  middle  fraction  is  the  expression  whose  limit  (or  rather  that  of  its 
reciprocal)  is  to  be  determined.  The  last  relation  shows  that  it  always  lies 
between 

cos^    and   -•, 

^  cos^ 

2 

the  first  being  a  proper,  the  second  an  improper  fraction.     If  h  approaches 

the  limit  zero,  both 

h         ,       1 
cos-    and 

^  cos^ 

2 

approach  the  limit  unity ;  and  the  ratio  which  is  always  intermediate  must 
also  approach  unity ;  all  three  of  the  quantities  thus  approach  the  same  limit. 


124  CALCULUS  [Ch.  IV 

That  is,  the  curve  cuts  the  a;-axis  at  angles  of  45°  and  135^ 
alternately.     The  equation 

sin  (x  +  2  nir)  —  sin  rr, 

shows  that  the  same  value  of  y  which  corresponds  to  any 
value  of  X  corresponds  also  to  the  value  x  +  ^lnir;  we  can 
therefore  construct  the  entire  curve  by  moving  repeatedly 
the  part  extending  from  0  to  2  7r,  a  distance  equal  to  27r 
either  to  the  right  or  left. 

The  curve  is  a  simple  periodic  curve^  and  is  called  the 
sine-curve. 

The  derivative  of  cos  x  is  obtained  in  a  manner  similar 
to  the  above.     We  have  * 


^Qx  cos  (a:  +  ^)  —  cos  a; 
{^6)    ~ 


o-     X  -{-  h  —  X       .     X  -\-  h  -{-  X 

2  sin  —— •  sui  —^ — ^i— 


=  -sm(x  +  '^^ 


h 
.    h 

2 


and  the  limit  of  this  expression  as   h  approaches   zero   is 
—  sin  X ;  the  derivative  of  cos  a;  is  —  sin  a; ;  that  is, 

(4)  -  ^^^«^— sina;. 

Art.  3.  Geometric  interpretation  of  the  sign  of  the  deriva- 
tive. It  is  of  interest  to  determine  the  signilicance  of  the 
negative  sign  in  the  last  equation.  We  know  that  the 
derivative  is  the  limiting  value  of  the  ratio  of  A  cos  x  to  Aa;, 
where   A  cos  x  indicates  the  increment  that  cos  x  receives 

*  Formula  43,  Appendix. 


2-3.]       DERIVATIVES   OF  THE   SIMPLER   FUNCTIONS        125 

when  X  increases  by  Ax.     This  increment  is  negative;  i.e. 
cos  X  at  first  decreases  when  the  arc  x  increases ;  *  and  as  a 


matter  of  fact  cos  0  =  1  and  cos 


TT 


0. 


The  above  statement  holds  for  every  function  whose 
derivative  is  negative ;  it  can  be  enunciated  in  the  form  of 
the  following  theorem : 

If  a  function  increases  continually/ for  a  sequence  of  increas- 
ing values  of  x,  its  derivative  for  these  values  of  x  is  positive  ; 
hut  if  on  the  other  hand^  it  decreases  continually^  its  derivative 
is  negative. 

This  fact  may  be  illustrated  in  the 
following  manner  :  Let 

(1)  i,=fix) 

be  a  function  whose  graphic  repre- 
sentation is  the  accompanying  curve 
(Fig.  40).     We  have  for  this  curve 


(2)  tanT=^=^^^. 

dx         dx 


Fig.  40. 


If  B  is  the  highest  and  D  the  lowest  point  of  the  curve, 
the  ordinate  (i.e.  the  function),  increases  from  A  to  B  and 
from  7>  to  U,  and  we  easily  see  that  along  these  portions  of 
the  curve  the  angle  r  is  acute  and  tan  r  is  positive  ;  on  the 
other  hand,  along  the  portion  of  the  curve  BCD.,  the  ordi- 
nate or  the  function  continuously  decreases  so  that  in  this 


*  This  is  tnie  as  long  as  the  arc  x  lies  between  0  and  tt.  If  aj  >  tt,  sin  x 
becomes  negative,  and  therefore  A  cos  a;,  positive  again,  etc.  Thus  equation 
(4)  agrees  completely  with  the  fact  that  in  the  first  quadrant  the  cosine 
diminishes  continually  from  unity  to  zero,  and  in  the  second  quadrant  from 
zero  to  minus  one,  while  in  the  third  quadrant  it  increases  from  minus  one 
to  zero,  and  in  the  fourth  quadrant  from  zero  to  plus  one. 


126  CALCULUS  [Ch.  IV. 

case  T  is  an  obtuse  angle,  and  tan  r  is  accordingly  negative, 
and  at  the  points  B  and  I)  the  tangent  is  parallel  to  the 
a:-axis  and  tan  r  is  zero. 

Exercise.  Construct  the  graph  for  y  =  cos  x  and  discuss  it  as  y  =  sin  x 
was  discussed  above.  Show  that  the  curve  of  Fig.  39  will  represent 
cosx  if  the  origin  be  shifted  ^  radians  along  the  x-axis,  O'Y'  in  the 
figure  being  taken  as  the  ^/-axis. 

Art.  4.  Derivatives  of  sums  and  differences.  If  f{x)  and 
^{x)  are  two  functions  whose  derivatives  are  known,  the 
derivative  of  their  sum  is  found  in  the  following  manner : 

We  form 

(1)       Ui^  +  ^) + "^(^  +  ^)]  -  [/(^) + v^y\ 

h 

-        h        +         h        ' 

the  limit  of  (1),  when  h  approaches  the  limit  zero,  is,  by 
definition,  the  derivative  sought ;  and  the  limits  of  the  frac- 
tions in  the  right  member  are  the  derivatives  of  f(x)  and 
^(x),  respectively;  we  have  then 

^^^  da^  ~    djc    ^    d3c    ' 

In  words,  the  derivative  of  the  sum  of  two  functions  is  equal 
to  the  sum  of  their  derivatives. 

It  is  apparent  that  this  holds  good  for  any  number  of 
functions,  or 

(3)  £[/(^)+K^)  +  t(^)+-] 

^dfjx-)   ^  d4>(x-)  ^  df(x)  ^ 
dx  dv  dx 


3-4.]        DERIVATIVES   OF  THE  SIMPLER   FUNCTIONS        127 

The  derivative  of  the  difference  of  two  functions  is  ob- 
tained in  a  similar  manner.     Here  we  form 

it 

_f(x  +  h}-  fix)      <l>(x+h)-  <t>(.^} 
~  h  A  ' 

and  allowing  h  to  approach  the  limit  zero,  we  have 

d[f(x}  -  0(a;)]  _  df(x}      d(l>(x) 
dx  dx  dx 

In  words,  the  derivative  of  the  difference  of  two  functions  is 
equal  to  the  difference  of  their  derivatives. 

For  brevity,  single  letters,  u,  v^  w^  •••  are  often  used  to 
denote  functions  of  x  instead  of  /(^),  </)(^),  '«/^(^)?  ••>  a-nd 
with  this  notation  the  above  results  may  be  stated  in  the 
more  compact  form, 

d(u  -\-  V  -\-  w  -\-  •")  _  du      dv      dw 
dx  dx      dx      dx 

d(u  —  v')  _du  _dv 
dx  dx      dx 


EXAMPLES 

,     d(x  +  sin  x)       (1       ,  d  sin  x      ,    , 

1.    — — ■ ^  =  —  X  -\ =  1  +  cos  X. 

dx  dx  dx 


2    d(x^  -  cos  x)  ^  d(x^)  (/(cos  ^)  ^  o  x  |  sin  x. 
dx                 dx  dx 

^    d(x^  +  x'^-x)  ^  d(x^)  ^  d(x^)      r/x  ^  3  ^2  I  o  -^.      i^ 
dx                  dx  dx        dx 

10 


128  CALCULUS  [Ch.  IV. 

Art.  5.   The  derivative  of  cf{x),  c  being  a  constant.     We 

form 
..  cf(,x  +  h^-  cfCx)  _    fix  +  A)  -f(x) 

(,1)  ^^—  -C  ^  , 

and  taking  the  limits  as  h  approaches  zero,  we  obtain 

^^  d^~^    dx    ' 

the  constant  is  thus  seen  to  become  a  factor  of  the  derivative. 
For  example,  the  derivatives  of 

ax^^    b  sin  x,   c  cos  x^ 

are  nax"~\   5  cos  a;,    —  c  sin  a;,  respectively. 

Art.  6.  The  derivative  of  a  constant.  What  is  the  deriva- 
tive of  a  function  that  is  known  always  to  have  the  same 
value  ?     Let 

be  a  function  such  that  for  all  the  values  of  the  variable  x, 
y  has  the  same  value ;  the  numerator  of  the  fraction 

f{x^}i)-f(x^ 
h 

is  then  equal  to  zero  for  every  value  of  x  ov  h\  the  fraction 
is  therefore  always  zero,  and  hence  its  limit  must  also  be 
zero,  and  we  have,  if  c  represent  a  constant, 

(1)  ^  =  0. 

doc 

In  words,  the  derivative  of  a  constant  is  zero. 
This  conclusion  can  be  illustrated  geometrically.     Inas- 
much as  the  function  y  may  be  represented  by  an  equation 
of  the  form 


5-6.]       DERIVATIVES   OF  THE  SIMPLER  FUNCTIONS        129 

the  curve  corresponding  to  this  equation  must  be  a  straight 
line  parallel  to  the  axis  of  abscissae,  and  consequently  the 
angle  which  it  makes  with  the  axis  of  x,  as  well  as  the  tan- 
gent of  this  angle,  must  be  equal  to  zero ;  that  is, 

^  =  0. 
dx 

EXAMPLES 

^    d(dx'^  +  ^x-l)^d(Dx^)   ^dCSx)      d(l)^.d(x'^)  ^   3^^^iQ^i3, 
dx  dx  dx  dx  dx  dx 

^    d(7  x3  -  6  x2  -f  4)  ^  d(7  x^)      d(Q  x^)  ^  g^  ^,      ^o  ^ 
dx  dx  dx 

^    dCax  +  bsh\  X  +  c  cosix)  ,   , 

3.  — ^^ ■ ^  =  a  +  b  cos  X  —  csmx. 

dx 

4.  li  y  =  ax^  +  hx'^~'^  +  cx^-^  +  •••  -{■  px"^  +  qx  ■{■  r,  wherein   a,  b,  c,  "• 
p,  q,  r  are  constant  quantities,  then 

^  =  nax""-^  +  (n  -  l)hx''-^  +  (n  -  2)c^«-3  +  ...  +  2px  +  q. 
dx 

EXERCISES  XIII 

Find  the  derivative  of  each  of  the  following  expressions : 
(In  this,  and  the  other  sets  of  exercises,  the  first  portion  of  the  exercises 
can  usually  be  solved  without  the  use  of  pencil  and  paper.  It  is  recom- 
mended that  this  be  done.  The  number  of  exercises  which  can  be  solved 
thus  will  vary  with  different  persons.  Recourse  should  always  be  had  to 
written  work  whenever  it  becomes  confusing  to  hold  the  computations 
in  the  mind.  On  review  it  should  be  possible  to  solve  a  large  number 
without  pencil  and  paper.) 


1.    x\ 

7.    -f. 

12.   ^bx\ 

18.  ^^ 

2.   x\ 

3 

13.  4^. 

n 

3.   3a:6. 

«•    'f/- 

c 

19.   x<'+\ 

4.   2x\ 

17 
9.   (a-6)x. 

14.  Qx^. 

15.  4  2-8. 

20.  {c-\rd)x'-^ 

21.  2< 

3 

5.   ^^ 
10 

10.  If 

16.   na;". 

6.    -x4. 

11.    ~ax\ 

17.   2x^, 

22.  x-b. 

130  CALCULUS  [Ch.  IV. 

23.  x'^+Sx.  29    a:^^  +  x^^  34.    sin  x  +  cos  x. 

12 

24.  2x2-5x  +  4.  35.   2  sin  x  -  5  cos  x. 

30.  a:«  +  ax. 

25.  5  x2  +  10  X  -  3.  36.   x  +  cos  x. 

31.  x""*"^ x""*. 

26.  x^  -  1.  *  *  37.    4  +  Cos  X. 

««    xg+  x^ 

27.  2x3  +  3x2.  *''^'   ~^^  38.   3x-5sinx. 

28.  x5  -  5  X.  33.  sin  x  -  2. 

39.  12  x2  -  12  sin  x  +  cos  x.  42.   ax^  -  4  ftx^  +  6  cx^  -  4  c?x  +  e. 

40.  ax^  +  6x2  _^  ex  +  d.  43.    m  cos  x  —  r  sin  x. 

41.  ax4  +  bx^  +  cx2  -f  rf:c  -f  e.  44.    (a  —  b)  cos  x  +  (6  -  a)  sin  x. 

Art.  7.  The  derivative  of  a  product.  To  obtain  the 
derivative  of  the  product  of  two  functions  f(x)  and  </)(ic), 
we  alter  the  quotient, 

h 

whose  limit  when  h  =  0  is,  by  definition,  the  derivative  which 
we  seek,  by  adding  and  subtracting  in  the  numerator  the 
quantity /(a:)  •  (/>(a:  +  A),  obtaining  thus 

h 
_/Cx  +  h)(l>(x  +  h')-f(x)4>(x  +  h)-^f(x)<l>(x-\-h)-fCx:}(t>(x) 

h 

If  we  now  allow  h  to  approach  the  limit  zero,  we  have 

or,  on  introducing  a  more  compact  notation, 
doc  doc        doc 


I 


6-7.]       DtJRlVATlVES   OF  THE  SIMPLER  FUNCTlOl^S        181 

In  words,  the  derivative  of  a  product  of  two  factors^  is  the 
first  factor  into  the  derivative  of  the  second^  plus  the  second 
factor  into  the  derivative  of  the  first. 

If  the  product  whose  derivative  is  to  be  determined  con- 
tains more  than  two  factors,  it  may  be  divided  up  in  some 
way  or  other  into  two  factors  before  differentiation. 

EXAMPLES 

^^   fj(x  sin  x)^^.^^dx^  ^d^nx  ^  ^^^  ^  +  ^  eos  x. 
dx  dx  dx 

n    ^Csin  a:  COS  a:)  f/ sin  a:  ,     .       d  cos  x  o  •   o 

2.   -^ ^  =  cosa:— h  sin  x =:  cos^  a:  —  sin^  a:. 

dx  dx  dx 

3.ii^E^^^^l^  =  a(2xcoBx-x^smx). 
dx 

^    l(£!sin.^±«cos^^3^2si,,^  +  ^3cosx-asma:. 
dx 

5.  Given  the  function  x'^  sin  x  cos  x,  we  find,  on  taking  x"^  as  one  factor 
and  sin  x  cos  x  as  the  other,  that 

dCx'^sm  X  cos  x^        .  d(x^^   ,     9r/Csin  a:  cos  x) 

— >^ ^  =  sm  X  cos  X -^ — ^  +  x^-i ^ 

dx  dx  dx 

=  2  X  sin  X  cos  x  +  .^^(cos^  x  —  sin^  x),  by  2. 

To  deduce  corresponding  formulae  for  the  case  of  three 
factors,  we  have 

dCuvw^  du  ,      d(vw)  du  ,      f    dv  ,      dw\ 

~-^ — ^  —  vw—--\-u  -'^- — -  =  vw-y{-u[w-~-\-v—-]'^ 
dx  dx  dx  dx         \    dx         dxj 

.1    .  .  d(uvw^  du  ,         dv   ,        dw 

that  is,  —^ — -^  =  vw  ■ — -\-  %iw \-  uv 

dx  dx  dx  dx 

We  observe  that  the  derivative  of  a  product  of  three 
factors  is  the  sum  of  the  derivatives  of  each  factor  multiplied 
by  the  other  two  factors.  It  is  easily  seen  similarly  that  the 
derivative  of  the  product  of  k  factors  is  the  sum  of  k  terms, 


132  CALCULUS  [Ch.  IV. 

each  of  which  consists  of  all  the  factors  save  one  multiplied 
by  the  derivative  of  that  one  factor,  the  derivative  of  any 
factor  occurring  in  one  and  only  one  term.  The  formal 
proof  may  be  supplied  by  the  student. 

EXERCISES    XIV 
Find  the  derivatives  of  the  following  expressions : 

1.  y  =(x  +  2)(x  —  ^).*  7.   y  =  cos3a:(=  cosoT'COs'^a:). 

2.  y  =  sin  x  cos  x.  8.   y  =  cos*  x. 

3.  y  =  sin^  x  (  =  sin  x  •  sin  x).  9.   y  z=  cos^  x  sin^a:. 

4.  y  =  x'^  cos  X.  10.   y  =  x^  cos^  x. 

5.  y  =(4a;2  +  l)(3a:;8  -  5).  11.   y  =  cos2a:(=  cos^a:  -  sin^a:). 

6.  y  =  cos^x.  12.   y=(a:2  +  l)(x3  +  2)(a:4  +  3). 

Art.  8.  The  derivative  of  a  quotient.  We  now  proceed 
to  deduce  the  derivative  of  the  quotient  of  two  functions. 
At  once  denoting  the  two  functions  by  u  and  v,  and  putting 

(1)  2^=-. 

we  get  u  =  7/v^ 

and  on  forming  the  derivative  of  both  of  its  members,  we 
find 

^ON  du  _    dy         dv 

dx         dx         dx 

from  which  the  required  value  of  -^  is 

dx 

(3)  ^  =  Y— -3/— Y 

dx      V  \dx         dx) 

*  Though  not  necessary,  it  is  often  convenient  to  use  a  single  letter  to 
denote  the  expression  to  be  differentiated.  Of  course,  y  is  simply  another 
name  for  the  expression  on  the  right  in  each  case. 


I 


7-8.]        DERIVATIVES   OF  THE  SIMPLER  FUNCTIONS        133 

On  substituting  the  value  of  7/  in  the  right-hand  member, 
we  have 


du         dv 

V u  — 

dy_ 

dx         dx 

dx 

v^ 

^,  du     ^,  dv 

d  fu\_ 

v~ u—— 

ddc         doo 

doc\v  I 

v^ 

or 
(4) 

In  words  :  the  derivative  of  the  quotient  of  tivo  functions  is 
the  denominator  into  the  derivative  of  the  7iumerator  minus  the 
numerator  into  the  derivative  of  the  denominator  all  divided  hy 
the  square  of  the  denominator.  The  key  word  denominator 
helps  to  make  the  above  easy  to  remember. 

I.    To  determine  the  derivative  of  tan  x  and  cot  x.     Since 

^rx  ■.  sin  rc 

(5)  tan  X  = » 

cos  a:: 

we  have  m  the  case  in  hand,  u  =  ^uix^  whence  — =  cos  a;, 

J  dx 

and  v  =  cos  x,  whence  --  =  —  sin  a; ;  then 
dx 

du        dv 

V- U-— 

dx        dx  _  cos^  x  +  sin2  x  _     1       * 
v^  cos^  X  cos'^  X 

or 

d  tan  a?         1 


(6) 


die        cos^a?' 
1 


i.e.  the  derivative  of  tan  x  is 

cos- a; 

Since 

^fTx  ,  COS  a: 

(7)  cota;  = 


sma^ 


*  Formula  28,  Appendix. 


134  CALCULUS  [Ch.  IV. 

in  this  case  u  =  cos  x^  and  v  =  sin  x,  and  accordingly, 

du         dv 

V u 

dx         dx      —  sin^  x  —  cos^  x  1 


v^  sin^  X  sin^  x 

d  cot  a?  1 


or    (8)  ,       -      .  V-' 

1 


i.e.  the  derivative  of  cot  x  is  — 


sin^  X 


II.  Further,  let  «/  =  -,  where  «  is  a  constant;  here  u  =  a, 

X 

1  ■,  du      p.    dv      ^ 

and  v  =  a; ;  and  -—  =  0,  — -  =  1; 

whence 

III.  Likewise, 

(10)  -fseca.  =  Af    M  =  _-sm^      sm:r 


c?a;  c?a;  Vcos  xj  cos^  a^       cos^  a; 

sin  X       1 


tan  a:  sec  x. 


cos  rr    cos  x 
Similarly,. . 

(11)  — *"cosec  x  =  —  cot  X  cosec  a;. 

dx  ♦  .  . 


dx\a  —  xJ 


(a  -  a.)  ^^^±^  -  (a  +  :r) '^^^^^ 


(a  —  0^)2  (a  —  xy 

dx^t       9  6^  sin  X 

sin  a;  —  —  x^ ^       .  o 

^      d  I   x^  \_  dx  dx     _1x  sm  a:  —  a:^  cos  rr 

c?a:\sina:jy  sin'^a;  ""  sin^a: 

VI.    According  to  Boyle's  Law  (p.  3),  we  have  for  the 
volume  V  corresponding  to  the  pressure  j9,  the  equation 


r 


8.]  DERIVATIVES  OF  THE  SIMPLER  FUNCTIONS        135 

where  p^  and  Vq  stand  for  the  initial  pressure  and  volume, 
respectively.     By  writing  this  equation  in  the  form 

P 
we  obtain  in  accordance  with  II, 

dp  p^ 

The  derivative  is  negative  ;  it  is  also  (p.  108)  the  limiting 
value  of  the  ratio  of  Av  to  Ajt?,  which  represent  the  incre- 
ments of  volume  and  of  pressure ;  the  negative  sign  indi- 
cates that  as  the  pressure  increases  the  volume  decreases 
(p.  125).  The  relation  between  the  decrease  in  volume  to 
the  increase  in  pressure  is  according  to  our  equation 
inversely  proportional  to  p^ ;  for 

i?  =  2,  3,4,  ... 

this  relation  is  proportional  to 

4'  9'   16'  '*•• 
If  the  pressure  be  greatly  increased,  the  decrease  of  volume 
soon  becomes  very  slight,  a  conclusion  in  perfect  accordance' 
with  the  experimental  observation  of  gases. 

EXERCISES   XV 
Find  the  derivatives  of  the  following  expressions : 

1     ^±i.  5    ^!_+3.  9,    a:2+4a:-2       ^^      \^ 

*    a;  -f  2  *    a;8  -  1  *   a:^-  4  a:  +  2  '   car" 


1  >;    ar  sin  a;  t^    sin  a:  +  tan  a:  14. -• 

6. -.  10.      .  1    _|_  a;10 

X  2  a:  — 3  cosa; 

15.    cos  X  cot  X. 

-4*  '''   -. ^——7>'  11-   ^tana:.  le        ''^eca: 

«'-«^  +  ^**  1+cosa: 

1_.  8  si"  ^  12     ——'  17    ^^  -  ^^  +  7_ 

5a;^  *   a  +  icosa:  a;"  '     hx^  —  \x^ 


136  CALCULUS  [Ch.  IV. 

Akt.  9.  Logarithmic  functions.  Our  conception  of  loga- 
ritlims  consists  in  regarding  all  numbers  as  powers  of  a 
fundamental  number,  the  base ;  the  exponent,  which  indi- 
cates what  power  of  the  base  equals  the  number  in  question, 
is  called  the  logarithm  of  the  latter. 

The  tables  of  logarithms  in  general  use  take  the  number 
10  as  base,  because  of  the  advantages  thus  obtained  in 
numerical  calculations  with  logarithms;  we  shall  see  later 
that  in  theoretic  mathematics  there  is  an  advantage  in  using 
a  system  of  logarithms  with  another  base.  In  what  follows 
immediately  we  leave  the  base  of  the  system  of  logarithms 
undetermined. 

In  order  to  obtain  the  logarithmic  derivative,  we  have 
to  find  the  limit,  when  h  approaches  the  limit  zero,  of  the 
quotient 

^       log(^  +  ^^)  -  log  ^  ^  1 ;     ^L±A*=  I  logfl  +  ^\ 
h  h  X  h         \        xj 

1 

(2)  =i"^K^+3T+ 

The  right  member  of  this  equation  is  not  in  a  form  which 
permits  its  limit  to  be  discerned  immediately,  but  requires 
a  somewhat  long  discussion.     We  put 

(B)  -  =  ^,  that  is,  ^  =  — > 

X      6  h     X 

and  then  have 

('-i)'-('-IT=!('-i  ' 


*  Formula  6,  Appendix.  t  Formula  8,  Appendix. 

I  Formula  2,  Appendix. 


9.]  DERIVATIVES   OF  THE  SIMPLER  FUNCTIONS        137 

Substituting  in  (2),  we  obtain  finally 

^^^  i2&(£±|).zi££f  =  liog(i  +  lJ. 

We  have  next  to  determine  the  limit  of 
log(l  +  |J,orof(l  +  lJ, 

when  h  approaches  the  limit  zero.  Equation  (3)  shows  that 
when  h  is  approaching  the  limit  zero,  S,  on  the  other  hand,  is 
continually  increasing;  we  have  then  to  find  the  limit  of 
the  above  expression  when  B  increases  without  bound. 
To  simplify  matters,  we  assume  at  first  that  S  is  always  a 
positive  integer.*  We  have  then  according  to  the  Binomial 
Theorem,! 


3 


^1^  1-2  ^         1.2-3  '       • 

Since  S  is  a  positive  integer,  the  right  member  contains 
8  +  1  terms. 

We  now  seek  to  find  the  limit  of  the  right  member  when  k 

approaches  the  limit  zero.     In  this  case  -  also  approaches 

0 

the  limit  zero,  and  by  inspection  of  the  right  member  of  (5), 


*  It  can  be  proved  that  all  the  following  conclusions  are  true,  even  with- 
out such  an  assumption,  which  we  make  only  to  render  our  treatment 
simpler. 

t  Formula  3,  Appendix. 


138  OALCULUS  [Ch.  IV. 

we  see  that  under   these  circumstances   it  approaches  the 
limit 

The  limit  of  the  left  member  of  (5)  is  designated  by  e ; 
that  is,  we  put 

(7)  '='+\+h+h+h+-' 

the  number  e  thus  defined  plays  just  as  important  a  role  in 
mathematics  as  does  the  number  tt. 

Like   TT,  it   can   be   calculated   only  approximately.     Its 
value  to  the  tenth  decimal  place  is 

^  =  2.7182818284. 

The  calculation  as  based  upon  the  above  equation  is  very 
simple.     We  find 


i.u. 

I-- 

1 

3! 

=  1:3  =  0.16667 

1 

4!' 

=  1:4  =  0,04167 

1 

5l'' 

=  1:5  =  0.00833 
4! 

1 

6! 

=  1:6  =  0.00139 

l  +  -  +  l-j-l+l  +  l  +  l=  2.71806  -, 
1     2!      3!      4!      5!      6! 

and  thus  obtain  the  first  three  decimal  places  exactly.  The 
series  is  accordingly  very  well  adapted  to  the  approximate 
calculation  of  the  actual  value  of  e. 


9.]  DERIVATIVES   OF  THE  SIMPLER   FUNCTIONS        139 

We  have  then 

lim    ,         I 


and  substituting  this  result  in  (4).,  we  find 

Thus  far  we  have  made  no  decision  as  to  what  base  of  our 
system  of  logarithms  we  are  to  adopt.  If  we  take  10  as 
the  base,  the  corresponding  value  of  log  e  is 

log^o  6  =  0.43429.-. 

The  derivative  assumes  the  simplest  form,  however,  when 

log  g  =  1 ; 

that  is,  when  the  system  of  logarithms  has  e  as  its  base. 

The  fundamental  notion  of  logarithms  as  applied  in  abbreviating 
numerical  calculations  was  first  formulated  and  published  by  Baron 
Napier  of  Merchiston  (in  Scotland)  in  his  Mirljici  Logarithmorum 
Canonis  Description  1614.  Though  Napier  himself  did  not  devise  loga- 
rithms to  the  base  e,  and  indeed  did  not  build  his  theory  upon  the 
notion  of  any  base,  yet  he  furnished  the  impetus  and  the  fundamental 
idea  which  speedily  led  to  the  setting  up  of  systems  of  logarithms  to  the 
base  e  as  well  as  to  the  base  10,  as  we  now  have  them,  and  accordingly, 
in  honor  of  this  great  invention,  logarithms  to  the  base  e  are  often  called 
Napierian  logarithms.  They  are  also  called  natural  logarithms,  because 
the  theory  of  many  problems  may  be  discussed  more  simply  when  these 
logarithms  are  employed. 

Logarithms  to  the  base  10  are  called  Briggean  logarithms,  in  honor 
of  Henry  Briggs,  a  contemporary  of  Napier,  who  proposed  this  base, 
and  also  common  logarithms,  because  they  are  used  almost  exclusively 
in  practical  computations. 

In  what  follows  we  shall  usually  employ  natural  loga- 
rithms. They  are  denoted  by  lognata;,  or,  more  briefly,  by 
logo;;    we  shall  use  the  briefer  symbol,  and  shall  always 


140  CALCULUS  [Ch.  lY. 

understand  by  log  x  the  natural  logarithm  of  x^  while  loga- 
rithms to  any  other  base,  as  a,  will  be  denoted  by  log^.* 
Referring    to   equation    (.8),   we   have   then   the   following 

formulae  : 

(9)  d\Q^oc  ^  1_^ 

doo        dc 

(10)  ^J^^  =  llog„e. 

Art.   10.    Relations    between    logarithms   with    different 

bases.     If 

a]^  =  X  and  If  =  x^ 

we  have,  in  accordance  with  the  definition  of  logarithms, 

m  =  log«a;,  r  =  log^a;. 
Further,  we  have  from 

a'"  =  h'\ 

by  taking  logarithms  to  the  base  a  on  each  side, 

m—  r  log„  h. 

By  substituting  in  this  equation  the  values  of  m  and  r,  we 
find 

l0ga^=  I0g6^l0g«^ 

or 

(1)  log.i.=.fe 

This  equation  furnishes  us  with  a  means  of  calculating 
the  logarithm  of  any  given  number  for  the  base  5,  when  we 
know  its  logarithm  for  the  base  a, 

*  The  notation  log  a:  is  that  usually  employed  by  English  and  American 
writers,  while  In  x  is  used  by  Continental  writers. 


9-10.]      DERIVATIVES   OF  THE  SIMPLER   FUNCTIONS        141 


In  particular,  if  a  =  10  and  b  =  e,  so  that  log«  x  repre- 
sents the  common  logarithm  and  log^a;  the  natural  logarithm, 
formula  (1)  passes  into 


(2) 


^°^"^=i!fS- 


Thus,  when  the  natural  logarithm  of  x  is  known,  we 
obtain  the  Briggean  logarithm  by  multiplying  the  former 
logarithm   by  a  constant 

-,  which  may 


Fig.  41. 


be  computed  once  for  all. 
It  is  known  as  the  modu- 
lus of  the  Briggean  loga- 
rithms, is  denoted  by  iHf, 
and  has  the  value  M= 
0.43429  .... 

In  conclusion,  we  give 
the  graphic  representa- 
tion of  the  natural  loga- 
rithm (Fig.  41)  ;  *  that  is,  of  the  equation 

(3)  ^  =  \og^x. 

We  obtain  the  following  table  of  corresponding  values  of 
x  and  ^,  as  well  as  of  tan  r, 

x=0,       —,    ■     -,       1,       e,       e2,       00, 

e^  e 

^  =  -  GO,     -  2,     -  1,     0,       1,       2,        00, 

tanT  =  oo,      e^,         e,        1,       -       -^,       0. 

e        e^ 


*  The  unit  of  length  consists  of  two  of  the  spaces  into  which  the  x-axis  is 
divided  in  the  figure. 


142  ,  CALCULUS  [Ch.  IV. 

As  there  are  no  logarithms  of  negative  numbers,  no  points 
of  the  curve  can  correspond  to  negative  values  of  x^  and  the 
accompanying  figure  is  the  graphic  representation  for  the 
logarithm  of  x.  It  shows  that  as  x  increases  from  1  to  oo 
the  logarithm  also  increases  to  go,  but  very  slowly,  while, 
on  the  other  hand,  as  x  decreases  from  1  to  0,  it  decreases 
very  rapidly  from  0  to  —  oo.  Moreover,  the  angle  which 
the  tangent  of  the  curve  makes  with  the  axis  of  abscissse, 
is  45°  at  the  point  where  x  =  1  and  y  =  0 ;  as  a;  diminishes 
from  this  point,  the  angle  increases  approaching  the  limit 
90°,  as  X  approaches  zero ;  as  x  increases  from  this  point,  the 
angle  approaches  zero,  as  x  grows  beyond  all  bounds.  The 
curve  intersects  the  axis  of  abscissae  at  an  angle  of  45°,  and 
it  has  the  ^-axis  as  an  asymptote. 

Art.  11.  Connection  between  ^  and  ~-  Suppose  we 
have 

(1)  y=fi^'), 

and  from  this,  expressing  x  in  terms  of  y, 

(2)  x  =  <l>{yy 

(3)  Then  ^  =    1^-,  /(^  +  ^i)-/(^) 

cA^  ^_  lim  <i>(iy^h^-^<iy^ 

^^  dy~h  =  ^  \ 

Here  h^  and  li^  may  be  quantities  of  any  form  provided 
they  can  be  made  to  approach  zero.    Accordingly,  we  choose 

or,  by  (1), 

(5)  •  A^=/(^  +  ^^)_^. 


10-^12.]      DERIVATIVES   OF  THE  SIMPLER   FUNCTIONS      143 

We  see  that  when  h^  approaches  zero,  h^  also  approaches 
zero,*  and  therefore,  by  (4)  and  (5), 

^^x  ^_    lim    <^[/(^  +  Ai)]-(^(y) 

W  dy-h,  =  0     f(x+h^)-f(x) 

Let  y^  be  the  result  of  substituting  x  +  h-^  for  x  in  equa- 
tion (1),  so  that 

But  since  equation  (2)  is  equivalent  to  (1),  equation  (2) 
must  be  satisfied  by  the  same  values  y-^ov  x  -^  \  which  sat- 
isfy (1),  and  we  have 
(8)  ^  +  A^  =  ^[/(^  +  A^)]. 

Now,  by  means  of  equations  (8)  and  (2),  equation  (6) 

becomes 

dx  __    lim  X  -\-  h-^^-—  X 


\^J 

dy 

lim                     \ 
K^^  fix  ^\)- fix) 

Comparing 

(9)  with  (3),  we  have,  finally, 

(10) 

doc       1 
dy     dy 
doc 

(11)  or 

dx     dy      ^ 
dy     dx 

Art.  12.   The  exponential  function.     From  the  equation 

(1)  X  =  log.y 
there  can  be  at  once  derived  the  equation 

(2)  y  =  e^. 

*  The  hypothesis  is  here,  as  always,  tacitly  made  that/(x)  is  a  continuous 
function  of  x  ;  see  pp.  160-165. 

U 


144  CALCULUS  [Ch.  IV. 

We  have  thus  obtained  a  new  function  6^,  which  may  be 
regarded  as  the  inverse  of  the  logarithmic  function.  This 
function,  i.e.^  the  a;th  power  of  e,  in  which  x  occurs  as  an 
exponent,  is  called  the  exponential  function. 

We  obtain  its  derivative  by  differentiating  equation  (1) 
with  respect  to  ^,  with  the  result 

dx  _\  ^ 

I   y 

or,  in  accordance  with  p.  143, 

dx      ^ 
Substituting  for  y  its  value  from  equation  (2),  we  have 

(3)  ^  =  e^. 

die 

In  words,  the  derivative  of  the  exponential  function  is  identi- 
cal with  the  function  itself. 

Art.  13.  Illustrative  discussion  of  the  exponential  func- 
tion. The  curve  of  Fig.  41  (p.  141)  may  be  regarded  as  the 
graphic  representation  of  the  exponential  function ;  for  from 
equation  ( 3 ),  p.  141,  we  have 

x  =  e^ 

and  if  in  this,  we  interchange  the  letters  x  and  i/, 

Hence,  by  supposing  Fig.  41  to  be  turned  so  that  the  positive 
portions  of  the  axes  are  interchanged,  we  obtain  the  graphic 
representation  of  the  exponential  function.  While  x  passes 
from  0  to  Qo,  e^  increases  with  great  rapidity  from  1  to  oo ; 
as  X  passes  from  0  to  —  oo,  the  exponential  function  decreases 


12-13.]       DERIVATIVES   OF  THE  SIMPLER  FUNCTIONS      145 

very  slowly  from  1  to  0.  For  every  positive  or  negative  value 
of  X  the  corresponding  value  of  ^  is  positive  in  accordance 
with  the  fact  that  e'-^  =1  :  e^.  The  same  is  true  of  tanr, 
inasmuch  as 

(1)  tanT  =  ^  =  ^-. 

ax 

The  significance  of  the  exponential  function  in  the  phe- 
nomena of  nature  is  illustrated  by  the  following  discussion. 
If  a  capital  of  e  dollars  be  invested  at  j9  %  interest  for  one 
year,  the  capital,  together  with  the  interest,  will  be  equal  to 

(2)  ^i  =  <'  +  <'4  =  <l  +  lfo> 

If  the  capital  e^  draw  interest  for  another  year  at  the 
same  rate,  the  sum  of  the  capital  and  in(;erest  at  the  end  of 
the  second  year  will  amount  to 

(3)  ''2  =  '^i  +  '^i-ifo  =  '^iO  +  ifo)  =  <l  +  ifoT- 

If  this  be  continued  for  n  years,  it  is  readily  seen  that  the 
capital  will  be 

If  we  now  suppose  that  the  interest  is  added  to  the  capital 
every  month,  and  thus  the  money  bearing  interest  is  in- 
creased every  month,  the  sum  of  the  capital  and  interest  at 
the  end  of  the  first  month  will  be 

""i  ""^  "^ ""' 10032  "  ""V"^  "^  iMoo} 

and  at  the  end  of  the  second  month  it  will  be 


146  CALCULUS  [Ch.  IV. 

and  so  on ;  in  a  year  the  value  of  the  capital  will  be 

(5)  "^^^'i^  +  vimf- 

It  is  easy  to  see  how  the  formula  will  become  altered  if 
the  interest  be  added  to  the  capital  every  day  or  every  hour. 
As  the  time  is  made  shorter  and  shorter,  we  approach  more 
nearly  to  what  actually  occurs  in  nature.  When  in  any 
process  of  nature,  an  active  force  increases  continuously 
from  its  own  action,  the  force  added  at  each  instant  imme- 
diately has  all  the  effects  and  powers  of  the  original  force, 
and  exerts  them  conjointly  with  it.  In  order  to  extend  the 
above  formula  to  the  case  when  the  interest  is  added  con- 
tinually to  the  working  or  interest-bearing  capital,  we  must 
substitute  for  the  number  12,  a  number  w,  that  increases  in- 
definitely ;  if  besides,  we  write  q  instead  of  j^,  and  let  G 

denote  the  amount  of  capital  and  interest  at  the  close  of  the 
year,  we  have 

^-{.'r.(i+l)-]- 


If  we  now  put 
we  have 


^  =  -,   or   71  =  hq, 
no 


and  we  find  finally,  by  taking  the  limit  of  this  expression  as 
n  (and  consequently  8)  increases  without  bound,  that 

(6)  C=ce^. 

The  exponential  function  e^  thus  determines  the  amount 
of  an  active  or  interest-bearing  capital  after  a  year's  time 

*  Fonnula  2, ,  Appendix. 


13-14.]      DERIVATIVES  OF  THE  SIMPLER  FUNCTIONS     147 

(or  any  other  given  unit  of  time),  it  being  assumed  that  the 
increase  at  any  instant  is  proportional  to  the  active  capital, 
the  number  q  being  the  proportion  of  increase  in  the  unit 
of  time,  provided  the  increase  is  not  added  to  the  active 
capital.* 

Art.  14.  Inverse  trigonometric  functions.     If 

(1)  x=  sin«/, 

y  may  also  be  regarded  as  a  function  of  a; ;  i/  is  an  angle 
whose  sine  is  x.     The  notation  isf 

(2)  1/  =  arc  (sin  x}  =  arc  sin  rr, 
which  indicates  that  2/  is  an  angle  whose  sine  is  x. 


*  This  result  is  due  to  James  Bernoulli  (1654-1705),  the  first  of  a  remark- 
able Swiss  family  group,  who  would  be  mathematicians  despite  all  obstacles. 
In  accordance  with  the  wishes  of  his  father,  James  studied  Divinity  ;  at  the 
same  time,  following  his  own  inclinations,  he  quietly  devoted  himself  to 
mathematics  and  astronomy.  Refusing  a  pastorate  which  was  offered  him 
after  the  completion  of  his  studies  and  travels,  he  settled  in  Basel  as  pro- 
fessor of  mathematics  and  physics.  Among  his  pupils  was  his  younger 
brother,  John  Bernoulli  (1667-1748),  for  whom  also  mathematics  had  irre- 
sistible attractions,  overthrowing  his  father's  plan,  which  destined  him  for  a 
commercial  career.  Upon  the  death  of  James  Bernoulli,  John  succeeded  to 
his  professorship,  and  held  it  until  his  own  death,  over  forty  years  later.  A 
nephew,  and  pupil  of  both  the  brothers  in  turn,  Nicholas  Bernoulli  (1687- 
1759),  professor  of  mathematics  at  Padua,  and  later  of  logic  at  Basel,  and 
two  sons  of  John  Bernoulli,  —  Nicholas  Bernoulli,  the  second  (1695-1726),  and 
Daniel  Bernoulli  (1700-1782),  both  for  a  time  members  of  the  newly  founded 
academy  at  St.  Petersburg,  —  complete  this  family  group  of  five  men,  who  for 
a  whole  century  were  prominent  figures  in  the  mathematical  world  and  leaders 
in  its  activities.  We  may  mention  also  a  third  son  of  John  Bernoulli,  viz. 
John  Bernoulli,  the  second  (1710-1790),  who  devoted  himself  to  physics; 
also  a  son  of  the  latter,  John  Bernoulli,  the  third  (1744-1807),  who  made 
some  contributions  to  the  history  of  mathematics. 

t  Many  English  and  American  writers  use  the  notation  sin-^ic,  etc.,  in- 
stead of  arc  sin  a;,  etc. 


148  CALCULUS  [Ch.  IV. 

In  a  similar  manner 

(3)  X  =  cos  3/, 

where  y  is  an  angle  whose  cosine  is  x,  yields 

(4)  ^  =  arc  (cos  a:)  =  arc  cos  x, 
and  likewise, 

(5)  a:  =  tan^,  a:  =  cot?/,  a;=sec^,  and  a:  =  cosec^ 
yield  the  inverse  functions, 

(6)  ^  =  arc  tan  x,    y  =  arc  cot  x^ 

y  =  arc  sec  x^     y  =  arc  cosec  x. 

These  functions  are  called  inverse  trigonometric  or  circular 
functions. 

We  obtain  their  derivatives  as  follows.  Differentiation 
of  equation  (1)  gives 

dx 

--=cos^, 

dy 


=  Vl-2:2,* 

or  by  p.  142,  ^=    ^^  ; 

dx     Vl  —  x'^ 
hence 

(7)  ^  arc  sin  a;  =        ^ 


In  the  same  way  equation  (3)  yields 
(8)  ^arccosi»  =  -        ^ 


^^  Vl  -  x^ 


*  Formula  29,  Appendix. 


14.]         DERIVATIVES   OF  THE  SIMPLER   FUNCTIONS        149 
From  the  first  of  equations  (5)  we  liave 
dx         1 


di/      cos^  2/ 


or,  since  cos^  ?/  = 


1  +  tan2  1/      l-\-x^' 


we  have  -—  =  1  -f  a;^^ 

dy 

dy  __1_ 


dx      1  +  a;2 


or 

or,  finally, 

(9)  ^arctaHa;  =  — L. 

doc  i  +  ic2 

Quite  similarly  we  find  from  a:  =  cot  y  (or  ?/  =  arc  cot  rr)  that 

(10)  -^  arc  cot  a;  = ^. 

die  1  +  i;c2 

EXERCISES  XVI 

j^     arc  cos  x 

1   -  X2    * 

,«     arc  tan  x 

13.   arc  tan  x  +  arc  cot  a:. 
€■'  14.    logf  a:  •  sin  x. 


Differentiate : 

1. 

a:  log  x. 

2. 

logo; 

a: 

3. 

log  x  •  e*. 

4. 

log  a: 

^e       COS  X 


X 


5.  (1  —  x2)  arc  sin  x. 

6.  (1  +  a:2)  arc  tan  a;.  log 

7.  6-(cosa:-sinaO.  ^^^   tan  a;  •  log  a:. 

8.  e'x\     Ans.  e^x\x  ^- ^).  17.   ?11L^ -f  log  a;. 

9.  e'(a:3-3a:2). 

18.   -i_+31oga:. 

10.  e^  arc  cot  a:.  log  x 


150  CALCULUS  [Ch.  IV. 

Art.  15.  Differentiation  of  functions  of  functions.  Our 
previous  formulse  enable  us  to  perform  the  differentiation 
of  a  large  variety  of  expressions,  but  they  do  not  suffice  to 
enable  us  to  find  the  derivative  of  such  simple  functions  as 

(1)  {a^  +  a^f,  sin  (x  -  a),  log  ^^^^, 

and  still  precisely  such  functions  as  these  occur  much  more 
frequently  than  do  sin  x^  a;",  or  log  x.  To  treat  these  func- 
tions, we  need  therefore  a  more  general  process. 

Each  of  the  functions  (1)  is  a  function  of  a  function  of  x, 
the  first  is  a  power  of  a^  +  x\  the  second  is  the  sine  of  a  dif- 
ference, the  third  is  the  logarithm  of  a  quotient.  They  have 
all,  therefore,  the  form 

(2)  y  =  F(u). 

where  u  is  itself  some  function  of  x^  which  we  denote  by 

(3)  u  =  <i>(x), 
so  that  we  may  write  y  in  the  form 

(4)  y  =  Fi<i>ix-)-]. 

From  equation  (2),  regarding  «/  as  a  function  of  i^,  we 
have 

lim    F(u-\-h^)-F(u}  _di/ 
^^>^  h,  =  0  h^  -  du 

This  is,  by  definition,  the  derivative  of  y  when  u  is 
regarded  as  the  independent  variable  (or,  more  briefly,  the 
derivative  of  y  with  respect  to  it),  provided  \  is  any  quan- 
tity which  may  approach  the  limit  zero. 


15.]         DJERlVATtvm  OP  THE  SIMPLER  J^UITCTIONS        161 

From  equation  (3),  we  have,  similarly, 

lim    <t>(x-{-  h^^-  (f)  (x}      du 


^"^                    k,  =  o              A, 

"dx 

From  equation  (4),  we  have 

.7.                 lim    F[<l>(x  +  hs):\- 

-F[4>(x)-]      dy 

<^'^              *siO                       h. 

dx 

Each  of  the  equations  (5),  (6),  and  (7)  defines  the  deriva- 
tive indicated  no  matter  what  expressions  are  used  as  ^j, 
Ag,  A31  provided  only  that  they  can  each  be  made  to  approach 
the  limit  zero. 

We  now  avail  ourselves  of  this  fact  to  make  a  special 
choice  of  the  quantities  A^,  h^,  h^,  viz. 

"^2  ^^  h'> 
h^  =  (\>  {x -\- h ^  —  (^  (x) , 

It  is  apparent  that  this  choice  is  permissible,  because  when 
^3  approaches  zero,  \  and  h^  evidently  approach  zero.* 
We  have  then  from  (5) 


dy^    lim    F[u  +  </>(a;  +  A3)-  ^(x)']  -F(u) 
du      h  =  ^  (f)(x-[-h^)-  <i>(x)  ' 

or  replacing  u  by  its  value  <i>Qc)^  we  have 
^  ^  du      h^^  <l>{x  +  A3) -  (l>{x) 


(9) 


du^    lim    <l>(^  +  Ag)  -  <f>(x) 
dx      h  =  ^  Ao 


*  The  dependence  of  our  results  upon  the  tacit  hypothesis  which  we  always 
make,  that  our  functions  are  all  continuous  functions  (p.  160),  is  here  very 
clearly  seen. 


152  CALCULUS  [Ch.  IV. 

But  we  have  identically,  for  all  values  of  h^  however 
small, 

<P{x  +  h^)-<\>(x)  A3 

_F[<i><ix+\^-\-Fl<\>(x-)-\ 
h 

Taking  the  limit  as  ^  =  0,  of  both  members  of  equation  (10), 
we  have,  comparing  with  equations  (8),  (9),  and  (7), 

(11)  ^.^  =  ^. 

da    dx     due 

We  have  thus  the  important  result : 

If  y  is  a  function  of  u^  and  u  in  turn  is  a  function  of  x^ 
then  the  derivative  of  y  with  respect  to  x  is  equal  to  the  deriva  - 
tive  of  y  with  respect  to  u^  multiplied  hy  the  derivative  of  u 
with  respect  to  x. 

We  proceed  at  once  to  apply  this  result,  by  finding  the 
derivatives  of  the  functions  given  in  (1). 


EXAMPLES 

1.   Let 

y 

=  {a2  +  x2}3. 

Put 

u  ■- 

=   «2  +  a;2^ 

and  accordin 

giy. 

y- 

=  u\ 

By  our  pr( 

2vious  ] 

results 

du_ 
dx' 

djL. 
du 

-  9  r  • 

-  Z  X  , 

=  3m2. 

Hence 

dy. 
dx' 

_  dy     du 
du     dx 

=  3M2.2a: 

=zQx{a'^  +  X 

2)2. 

15.]         DERIVATIVES   OF  THE  SIMPLER  FUNCTIONS        153 

2.   Let 

Put 
Then 


Therefore 

3. 

Put 
Then 


and 
Hence, 


y  = 

-  sin  (x  —  a). 

u  = 

-  X  —a. 

y  = 

--  sin  M. 

\ 

du_ 
dx 
dy_^ 
du 

=  1; 

:  COS  U. 

dy^ 
dx 

_  dy     du 
du     dx 

- 

-  cosu  =  cos  (a; 

-a). 

y  = 

1      a  —  X 

.log- 

b  —  X 

u  = 

a  —  X 

b-  X 

y  = 

--  log  M. 

du_ 

Ah-x)(-i)- 

-{a- 

x){ 

-1) 

dx 

(b- 

■  xy 

= 

a-b 
'  ib-x)^ 

dy_ 
du 

.1 

u 

■■ 

dy. 

1       a-b 

dx 

u 

(6- 

-xy 

b- 

-  X 

a  - 

ib- 

-b 

a  - 

-  X 

xy 

a 

-b 

{a-x){b-x) 

4.   To  find  the  derivative  of  a*. 

Put 

y-a-.     ^-.-7      '^r^"^\ 

We  have 

a  =  ei«g«; 

Put 

a;  log  a  =  u.     Hence  y  =  e«,  and 

iy^djidu 
dx      du     dx 

—  e**  •  log  a. 

Therefore 

^^-  =  a- log  a. 
dx 

154  CALCULUS  [Ch.  IV. 

The  work  of  computation  in  examples  like  the  preceding 
can  be  somewhat  abbreviated  by  not  formally  introducing 
the  function  u.  Thus  in  the  first  example  we  may  write 
at  once,  i 

dx~  ^     ^^  dx        ' 

where  all  that  remains  to  be  done  is  to  differentiate  the 
second  factor,  giving 

^  =  3(a2  +  ^2)2  .  2  a^  =  6  x(a^  +  x^y. 
dx 

Similarly,  the  otlier  examples  can  be  worked  without  the 
formal  introduction  of  m,  and  as  the  student  becomes  familiar 
with  the  method  of  this  section  by  practice,  he  will  find  that 
the  explicit  use  of  the  function  u  may  gradually  be  omitted. 
But  the  beginner  is  earnestly  advised  always  to  use  the 
auxiliary  function  u  until  he  has  acquired  thorough  control 
of  the  practical  application  of  the  method.  Even  then  he 
should  make  formal  use  of  the  auxiliary  functions  whenever 
the  expression  to  be  differentiated  is  complicated,  as  con- 
fusion and  errors  may  otherwise  easily  arise. 

The  results  of  this  section  can  be  extended  immediately 
to  functions  of 'functions  of  functions,  and  so  on.  In  such 
cases  several  auxiliary  functions  are  necessary  and  their 
formal  use  is  imperative. 


EXERCISES  XVII 

1. 

y=(^x-^y. 

7. 

y  =  log  x\ 

12. 

y  =  <^. 

2. 

.y={^x^-^y. 

8. 

y  =  \og^x. 

13. 

y  =  e^'-^. 

3. 

4. 

y  =  sin  5  X. 

y  —  Q,os,(2x^  —  3 

X). 

9. 

^  =  '°^4i- 

14. 
15. 

y  =  log  sin  x. 
y  —  sin  x'^. 

5. 

y={ax  +  hY. 

10. 

y  =  e^\ 

16. 

y  =  sin*  a:. 

6. 

y  =  \og2x. 

11. 

y  =  e'^+\ 

17. 

y  =  sin  4  x. 

15-16.]     DERIVATIVES   OF  THE  SIMPLER   FUNCTIONS      155 

18.  ?j  =  sin^  X  cos^  X.        21.  ?/ =  et^nx-i^  24.    ?/ =  arc  sin  e^ 

19.  ij  =  tan  xK  22.    y  =  arc  sin  x^.  gS.   y  =  tan  ^^  +  ^ 

20.  y  =  e«*"*.  23.  ^  =(arc  sin  xy.  "^  ~" 

Art.  16.  The  derivative  of  a  power  with  any  exponent. 

Considering  a;"  (^n  being  a  positive  integer),  we  have  shown 

that  — rr"  =  nx'^~^;  and  we  shall  now  show  that  —  =  nx^~^, 
dx  ax 

even  when  ?^  is  a  positive  fraction. 
Let  y  =  a:", 

and  let  n  =  -^ 

where  j9  and  q  are  positive  integers;  accordingly 

(1)  y  =  xl 

Raising  (1)  to  the  ^th  power,  we  have 

y"^  =  xP. 
Put  (2)  y'^  =  u, 

then  (p.  152),  du  ^du  dy^ 

dx      dy  dx 

(3)  =,f-^% 


From  (2) 
) 

Equating  (3)  and  (4),  and  solving  for 


ri\  ^^      d(xP} 


dx        dx 

dy^ 

dx 

dy  _  pxP~'^ 

dx     qy^'^ 

=  ^^^-i.0J)-^/+\by  (1) 

V    ^-1 
=  -xi    ' 

9 


156  CALCULUS  [Ch.  IV. 

P 
Therefore,  replacing  -  by  its  equal  n, 

(5)  — x"  =nx'^~'^. 
ax 

The  formula  for  the  derivative  of  a  power  holds,  therefore, 
as  well  when  9^  is  a  positive  fraction  as  when  it  is  a  positive 
integer.  We  show  further  that  it  holds  when  n  is  sl  negative 
integer  or  a  negative  fraction. 

Consider  ^  =  a;",  where  n=  —  r^  and  r  is  a  positive  integer 
or  fraction. 

Then  «/  =  — • 

^     x"- 

Applying  the  rule  for  the  differentiation  of  a  fraction,  we 

have 

dy 
dx 
Replacing  —  r  by  its  equal  n,  and  y  by  a;% 

(6)  ^  =  nx^-\ 

dx 

We  thus  have  the  general  result  that  the  formula  just 

written  is  true  for  all  integral  and  fractional  values  of  n. 

In  particular,  we  have 


(8)  —  V^  =  —x^  =  l-x~^  =     ^    - 

dx  dx  ^  ^-\/x 

These  formulse  are  frequently  applied. 


EXAMPLES 

T      rf    3/-2      „    -1        2 
dx  3-^x 

2    —1-\  —  JL(     -\\  —  —      -2  —  _SL 
dx\xl     dx  x^ 


18.]  DERIVATIVES   OF  THE  SIMPLER  FUNCTIONS        159 

EXERCISES  XVIII   (MISCELLANEOUS) 


Differentiate : 

1.  y  =  x^.  Q    y  =  x^.  ^^    //  =  sin2x. 

2.  y  =  5x^  „  ,.-  ^ 

1  T'  y  =  V  x^.  11.   y  =  Vsin  x. 

3.  y  =  x\  1 

^.   y  =  ax^  +  bx\  ^'    y=x  12.    y=x  cos:^:. 

5.   y  =  a  -\-  bx  +  cx^.  9.   y  =  6  x~^.  13.   y  =  x  log  x. 

1/1  11  ,      a;  16.    y=(x  -\-  l)(x  +  2). 

14.  y  =  x^  tan  a:  H *^      ^  y  v    ^^    y 

cosx  17.   y  =  (2a:2-4)(3a:  +  5). 

15.  y  =  v'l  — 'a,-'^.  18.   y  =  x^e'^. 

19.  y  =  sin"»:r  cos*'a:.         23.    w  =  e«*.  27.    ?/  =  log(l  +  x^) 

20.  y  =  cos  log  x.  24.   y  =  sec  (3  a:  +  5).        28.   y  =  log  =— t— • 

1  —  a: 

21.  y  =  log  sin"  x.  25.    7/  =(x2-14x  +  2)3.      ^9^   ^^  =  :r«  +  nx«-i. 

22.  y  =  e''^''.  26.    ?/  =  xWl  +  a:^.  30.   y  =  log  (log  a:). 

31.  y  =  logsin(aa:  +  ^>).  4O.   y  =  \og(x+Vl  +  x^). 

32.  2,  =  log  tan -^.  41^   ^  ^  ^x. 

33.  y=(x-^a)(x  +  b)(x+c).  *2.  y  =  e-  •  :i- 

34.  y={x-lXx-2)(x-^)(x-i).  ^^^  2/ =  tana;  +  1  tan^x. 

35.  y  =  (:.  -  a,){x  -  a,)...(x  -  o„).  ^^'  V  =  ^^'^  ^  ^^  "  -^• 

36.  y  =  e-in'^'l  45.  y  =  log^^i^^ 

37.  ?/  =  e^rctanx. 

,^.  46.   y={\ogx^y. 

38.  y  =  6"'^  _  .]_:,' 


39.   y  —  e«*(a  sin  x  —  cos  a:). 

48.  log  j!L221iiLrMM. 

\  a  cos  a:  -}-  6  sm  a: 

49.  z/  =  x^". 


47.   2/ 


arc  sm 

■l+a:2 

Ans. 

a6 

a2cos2x 

-62 

sin^a: 

Ans. 

a..«+n-l^^  log 

x^\) 

A  „o 

1 

50.  ,  =  lo8V^^±5.  .™.-^. 

^  Vl  +  a:2  -  a;  vTT^ 

51.  y  =  (sin  xy.  A  ns.  (sin  xy  {log  sin  a:  +  a;  cot  x] 


.    y=f-j.  Ans.   f-j  (loga -loga:- 1). 


52 

Vary 

12 


160  CALCULUS  [Ch.  IV. 

Hint—  Put  1^:^=  z.   Ans.  ^=  -  U^-^^  '  Vl  +  logl^]- 
X  dx         x^\    X    J  \-  X    J 

55.  y  = ^^'  Ans.  ^  = ^1 .« 

{9x-13yV9x^  -rSx  dx  (^  x -Vdyy/Qx^ -l)ix 

108-18V^-3x--^V^  ,    ■  ,„ 

c/-  ^  A        dy  \0x 

56.  ?/  = ^ ^725.       ^  — 


|+20xB^..o.,9  '  ,        ^V  27 

57.  ^= Ans.     '^ - 


58.   3/ 


V(8  +  5x6)3  (/x  x4\/(8  +  5x6)5 

1  1 


\\         x^J  \\     -  xV  dx       ^(x3+l)ii 


Art.  19.  Continuity  and  discontinuity.  We  reached  the 
notion  of  the  derivative  by  taking  up  the  problem,  to  deter- 
mine the  position  of  the  tangent  of  a  curve,  the  speed  of  a 
moving  point,  or  tlie  coefficient  of  expansion  of  a  metal  rod. 
In  each  of  these  cases  we  had  under  consideration  the  deter- 
mination of  a  quantity  with  a  definite  geometric  or  physical 
signification.  We  then  extended  the  method  of  computation 
of  the  derivative  which  we  used  in  the  instances  just  named 
to  the  more  usual  classes  of  functions  which  occur  in  mathe- 
matics. The  examples  which  we  have  treated  show  that  for 
all  these  functions  the  derivative  exists  and  may  be  deter- 
mined in  a  simple  manner,  and  besides  may  frequently  be 
brought  into  connection  with  some  natural  phenomenon. 

We  do  not  wish,  however,  to  pass  on  without  alluding  to 
the  fact  that  for  certain  functions  of  pure  mathematics,  as 


18-19.]     DEBIVATIVES   OF   THE   SIMPLER   FUNCTIONS      161 

well  as  in  some  applications,  exceptions  to  our  previous 
results  may  present  themselves. 

We  have  already  called  a  function  that  is  altered  little  at 
will  when  the  independent  variable  undergoes  a  sufficiently 
slight  change,  a  continuous  function.  (It  is  hardly  neces- 
sary to  remark  that  the  course  of  the  processes  occurring  in 
nature,  whether  physical  phenomena  or  cliemical  reactions, 
can  usually  be  represented  by  continuous  functions.  Natura 
nonfacit  saltus.^ 

We  say  of  a  curve  whose  equation  is 

and  which  has  a  break  in  its  course,  like  that  in  Fig.  42, 
that  it  is  discontinuous  for  the  value  t  =  OQ^  and  that  it  has 
a  discontinuity  at  this  point.  The  same 
expressions  are  also  applied  to  the  func- 
tion /(O  itself.  If  we  let  t  increase 
by  an  increment,  as  small  as  we  please, 
added  to  the  value  t  =  OQ^  f(t)  does 
not  change  (as  did  all  the  functions  _ 
hitherto  considered)  by  an  increment 
which  is  also  very  small,  but  by  an  in- 
crement at  least  equal  to  PP\  no  matter  how  small  the 
increment  of  t  may  be  taken.  The  derivative  has  two  values 
(usually  different)  at  the  point  t  —  OQ  which  correspond  to 
the  positions  of  the  tangents  (Fig.  42)  which  can  be  drawn 
to  the  curve  at  the  distinct  points.  P  and  P' . 

We  also  say  that  a  function  is  discontinuous  for  a  certain 
value  of  x^  when  for  that  value  of  x  the  function  becomes 
infinite.     Such  a  function  is,  for  example, 

1 

y  = 

a  —  X 


?' 


Q 

Fig.  42. 


162  CALCULUS  [Ch.  IV. 

As  ir  =  a,  y  becomes  large  without  bound  ;  the  function 
y  is  therefore  said  to  be  discontinuous  at  the  point  x  =  a. 
The  derivative  of  this  function  has  the  value, 

dy_       1 


dx      (a  —  x^' 

and  is  likewise  infinite  for  the  value  x=  a.  We  have  met 
several  such  functions  in  the  preceding  chapters.  To  select 
them,  we  have  only  to  examine  the  graph  and  see  whether 
the  curve  contains  branches  in  which  the  ordinate  becomes 
infinite  for  a  finite  value  of  x.  Such  curves  are,  for  example, 
those  for  log  x  (p.  141),  for  Boyle's  Law  (p.  4),  and  others. 

We  add  a  simple  example  of  a  function  which  becomes 
discontinuous  without  becoming  infinite.  For  brevity,  we 
introduce  temporarily  the  notation  I^  to  denote  the  greatest 
integer  contained  in  the  value  of  x  taken  positively.  Thus, 
74.5  =  4,  and  7_6.7  =  6. 

Considering  now  the  function 

y  =  x-{- 1^ 
we  see  that  when  —  1  <  a:  <  1, 

__  y  =  x-\-0, 

but  when  1  <  a:  <  2, 

_  y  =  x^-l, 

and  when  2  ^  a:  <  3, 

y  =  x  +  2^  etc.,  etc. 

When  x  approaches  1,  y  also  approaches  1 ;  z.e., 
lim 


Ti\_x  +  i;]=i, 


but  when  x  =1,  ^  =  2.  When  x  =  1,  any  diminution  in  the 
value  of  rr,  no  matter  how  slight,  causes  the  value  of  y  to 
diminish  by  more  than  a  unit ;  i.e.  y  is  discontinuous  at  the 
point  x=  1.     Likewise,  y  is  discontinuous  for  all  integral 


19.]         DERIVATIVES  OF  THE  SIMPLER  FUNCTIONS       163 


Fig.  43. 


values  of  x^   except    zero.      The   graph    for   this    function 
(Fig.  43)  has  a  break  at  each  integral  value  of  the  abscissa. 

A  second  illustration 
is  the  function  t/  =  I^. 
We  give  its  graph 
(Fig.  44),  and  leave 
the  detailed  discussion 
as  an  exercise  for  the 
reader. 

In  other  cases,  curves 
may  indeed  be  free  from 
breaks  or  discontinui- 
ties, and  yet  at  some 
point  have  a  sudden 
change     of    direction. 

In  this  case,  the  curve  for  the  derivative  has  a  discontinuity 
at  that  point.     A  good  example  of  this  is  the  curve  for  the 

vapor-tension*  of  a  sub- 
stance.    The  vapor  ten- 

F  sion  j9  of  a  solid  increases 

gradually    under     rising 
temperature ;   as  soon  as 

- — • «     the     melting     point     is 

reached  and  the  solid 
assumes  the  liquid  condi- 
tion, the  vapor-tension 
instantly  begins  to  in- 
crease at  a  more  rapid  rate,  and  the  vapor-tension  curve 
accordingly  has  a  sudden  change  of  direction,  as  is  illus- 

*  Every  solid  and  liquid  has  a  tendency  to  evaporate,  and  when  it  is  con- 
fined in  a  limited  space,  the  vapor  produced  exerts  a  tension  which  can  be 
measured  and  is  known  as  the  vapor-tension  of  the  substance. 


J K  2 

G H    1 

-3     -2     -1 


0     B 


Fig.  44. 


164 


CALCULUS 


[Ch.  IV. 


Fig.  45. 


trated  graphically  in  Fig.  45.  The  vapor-tension  itself  has 
no  discontinuity  at  the  melting  point,  for  at  that  tem- 
perature the  value  of  the  vapor- 
tension  is  just  the  same  for  the 
liquid  as  for  the  solid  body. 
But  the  derivative  is  discon- 
tinuous there ;  for  if  we  raise 
the  melting  temperature  t^OQ^ 
by  an  increment  A^,  as  small 
as  we  please,  the  derivative 
passes  from  the  value  of  the 
tangent  of  the  angle  a  to  the  tangent  of  the  angle  a' ;  the 
change  in  magnitude  of  the  derivative  does  not  approach 
zero,  no  matter  how  small  the  increment  of  temperature  is 
taken. 

We  conclude  with  a  formal  analytic  definition  of  con- 
tinuity, embodying  in  mathematical  symbols  the  ideas  we 
have  just  set  forth. 

A  function  /  (^x)  is  continuous  for  the  value  x=  a^if 

A"o  [/(«  +  ^) -/(«)]  =  0- 

The  function  f  (x)  is  always  continuous  if  we  have,  irrespec- 
tive of  the  value  of  x, 


A'ro[/(^+A)-/w]=o- 

In  a  continuous  function  this  limit  is  zero,  both  (1)  when 
h  approaches  zero  through  positive  values,  and  (2)  when  h 
approaches  zero  through  negative  values. 

In  a  discontinuous  function,  on  the  other  hand,  this  limit 
is  different  from  zero  in  at  least  one  of  these  cases. 
•  Thus,  in  the  function /(a^)=  x  +  /,.,  discussed  above, 


19.J         BE^VATIVES   OF  THE  SIMPLER  FUNCTIONS        165 


«) 


/ro[/(«+'^) -/(«)] 


(^a  being  a  positive  integer)  is  zero  in  case  (1),  but  unity  in 
case  (2).  If  a  is  a  negative  integer,  the  limit  is  —  1  in  case 
(1)  and  zero  in  case  (2).  If  a  is  zero,  or  not  an  integer, 
the  limit  is  zero  in  both  cases,  and  the  function  is  continuous 
for  these  values  of  a. 

In  what  is  to  follow,  just  as  in  what  has  preceded,  we 
shall,  as  a  rule,  pay  no  attention  to  the  possibility  of  excep- 
tional values  of  x^  for  which  the  function  is  not  continuous. 
The  results  which  we  deduce  in  this  manner  are,  of  course, 
only  proved  to  hold  for  continuous  functions,  but  we  shall 
usually  apply  our  results  to  functions  of  such  simple  nature 
that  we  shall  take  their  continuity  for  granted. 


/i 


CHAPTER  V 

THE  FUNDAMENTAL  CONCEPTIONS   OF  THE  INTEGRAL 

CALCULUS 

Art.  1.  The  problems  of  the  Integral  Calculus.  We 
learned  in  Chapter  III.  that,  in  the  theoretic  study  of  natural 
phenomena,  two  essentially  different  problems  confront  us  : 
one  assumes  the  laws  to  be  given  and  endeavors  to  find  out 
what  the  condition  of  the  process  is  at  any  moment;  the 
other  requires  the  deduction  of  the  law  controlling  the  entire 
process  from  the  facts  relating  to  the  single  phases  of  it;  the 
one  problem  is  the  inverse  of  the  other.  The  former  problem 
led  us  to  the  conceptions  and  methods  of  the  Differential 
Calculus.  We  now  take  up  the  inverse  problem,  and  begin 
with  the  following  illustrative  example  concerning  the 
motion  of  a  freely  falling  body  : 

The  value  of  v  at  any  instant  being  given  by  the  equation 

(1)  v  =  gt, 

to  derive  the  formula  for  the  space  I  traversed  in  the  interval 
of  time  t,  viz, 

(2)  l=^gtK 

We  might  start  out  from  the  equation  < 

dl 

V  =  ~^ 
dt 

which  we  have  already  established  (p.  109).    But  we  proceed 
in  a  somewhat  more  detailed  manner  in  order  to  make  per- 

166 


1.]     FUNDAMENTAL   CONCBPTtONS  OP  iNTEQRATtON     167 

fectly  clear  the  general  nature  of  the  process  for  use  in  other 
problems. 

Formula  (1)  was  obtained  by  means  of  an  auxiliary  point 
which  passes  at  a  uniform  rate  over  the  space  l^  —  l  during  the 
interval  of  time  t-^  —  t  —  T^  and  whose  velocity  by  definition  is 

(3)  v=:hj:zl. 

T 

If  now  we  take  the  limit  of  both  members  of  (3)  as  t 
approaches  zero,  we  have,  as  the  limit  of  F",  the  velocity  v  at 
Pj ;  but  the  limit  of  the  right  member  is  what  we  have 
called  the  derivative,  and  we  have  introduced  a  notation 
according  to  which  the  limit  of  the  right  member  is  denoted 

by  — -    As  its  value  pi'oves  to  be  gt,  we  have  finally 

(4)  .=!=,.. 

The  specifically  physical  part  of  the  problem  concludes  with 
the  establishing  of  equation  (4).  To  pass  from  (4)  to  (2)  is 
a  matter  of  pure  calculation  alone  :  a  function  I  is  to  be 
found  whose  derivative  is  known.  This  is  the  inverse  of  the 
problems  treated  in  Chapters  III.  and  IV.  There  we  had 
to  determine  the  derivatives  of  given  functions;  here  the 
derivative  is  given,  and  the  function  of  which  it  is  the 
derivative  is  to  be  determined.  In  the  example  under  dis- 
cussion, it  is  to  be  shown  that  I  =  ^  gt^  is  the  function  of 
which  gt  is  the  derivative. 

As  a  second  example  we  take  the  inversion  of  sugar  (i.e. 
decomposition  of  cane  sugar  in  aqueous  solution  into  dex- 
trose and  levulose  through  presence  of  acids).  By  speed 
of  reaction  we  mean  the  rate  at  which  the  sugar  is  in- 
verted.    If    as    much   sugar    is    added   to   the   solution   as 


168  ,  CALCULUS  [Ch.  V. 

disappears  on  account  of  the  inversion,  equal  quantities  of 
sugar  will  be  inverted  in  equal  intervals  of  time ;  the  reac- 
tion takes  place  with  constant  speed.  The  so-called  Law  of 
Mass  Action  states  with  reference  to  the  process  of  sugar 
inversion  that  the  mass  of  sugar  thus  inverted  in  the  unit 
of  time  is  directly  proportional  to  the  mass  of  sugar  still 
unchanged. 

Let  a  represent  the  mass  of  sugar  present  in  the  solution 
at  the  beginning  of  the  inversion,  and  let  x  be  the  mass  of 
sugar  inverted  in  the  time  t ;  then  the  mass  of  sugar  at  the 
close  of  the  time  ^  is  a  —  x.  Let  the  mass  of  sugar  inverted 
in  the  interval  of  time  A^  immediately  following  be  denoted 
by  Aa; ;  in  accordance  with  our  method,  the  inversion  is  to  be 
regarded  as  a  perfectly  uniform  process  during  the  time  A^ 
According  to  the  Law  of  Mass  Action,  the  amount  inverted 
is  proportional  to  the  mass  of  sugar  remaining  unaltered  in 
the  solution ;  if  its  value  for  the  unit-mass  of  sugar  be 
denoted  by  k^  then  its  value  for  the  mass  of  sugar  a  —  a;  is 
hia  —  :r),  which  expression  gives  the  mass  of  sugar  inverted 
uniformly  in  the  unit  of  time. 

If  a  —  xh^  the  amount  of  sugar  left  at  the  beginning  of 
the  interval  A^,  and  a  —  x^  that  at  its  end,  then  a  —  x>a—Xy, 
and  the  smaller  the  interval  A^,  the  more  nearly  the  quan- 
tities a  —  X  and  a  —  x^  are  equal.  The  amount  of  sugar 
inverted  in  the  time  A^,  if  the  mass  of  sugar  remains  constant 
as  at  the  beginning  of  the  interval,  would  be  k(a  —  x^At. 
But  this  is  too  great,  as  the-  mass  of  sugar  is  decreasing. 
If  the  mass  of  sugar  were  constantly  the  same  as  at 
the  end  of  the  interval,  the  amount  inverted  would  be 
kQa  —  x^M.  But  this  is  too  small,  as  the  mass  at  the  end 
is  the  smallest  mass  in  the  interval.  The  true  mass  in- 
verted is  then  less  than  the  former  and  greater  than  the 


1-2.]    FUNDAMENTAL  CONCEPTIONS  OF  INTEGRATION     169 

latter  mass.  If  we  denote  it  by  JcmAt^  then  m  is  less 
than  a  —  X  and  greater  than  a  —  Xy  Letting  m  =  a  —  x  —  €, 
where  0  <€<x^  —  x^  the  true  mass  inverted  is  denoted  by 

Jc(^a  —  X  —  e)At, 

But  we  have  previously  called  this  same  amount  of  sugar 

Aa:,  so  that 

Ax  =  k(^a  —  X  —  €)A^, 

A^  1 

or 


'.    Ax      k(a  —  X  —  e) 

Taking  the  limit,  as  Ax  approaches  zero  (then  x  approaches 
iTj,  and  e  also  approaches  zero),  we  have 

dt  ^        1 
dx     k(^a  —  x} 

The  chemical  law  has  thus  been  brought  into  a  mathe- 
matical form.  We  have  now  to  determine  what  functional 
relation  exists  between  x  and  t  that  finds  its  expression  in 
the  last  equation  ;  in  other  words,  what  function  of  x  has 

as  its  derivative.     It  is  easily  verified  by  differ- 

k(^a  —  x} 

entiation  that 

^  =  -  -  log(a  -  x) 

has  the  given  derivative,*  and  this  expression  is  the  func- 
tional relationship  between  x  and  t ;  it  may  also  be  written 

t=-log- 


k        a  —  X 

Art.  2.    Integrals.     The  preceding   illustrations  lead  us 
to  the  general  problem  of  the  determination  of  a  function 

*  A  fuller  discussion  of  this  step  will  be  given  on  pp.  181-182  and  189-190. 


170  CALCULUS  [Ch.  V. 

whose  derivative  is  known,  the  inverse  problem  to  that  dealt 
with  in  the  Differential  Calculus,  where  it  was  required  to 
find  the  derivative  when  the  function  was  given. 

The  branch  of  Mathematics  that  deals  with  the  finding  of 
functions  whose  derivatives  are  known  is  called  Integral 
Calculus.  This  name  will  immediately  seem  more  apt  if  we 
note  that  the  problems  treated  in  the  Integral  Calculus  seek 
to  find  the  laws  that  regulate  the  entire  (^integer)  course  of 
some  variation. 

Let  x^  be  the  derivative  of  a  certain  function  ^  ;  then 

m    ■  |=- 

Or,  more  generally,  let  the  derivative  be  denoted  by  f(x} 
and  the  required  function  by  F(^x)  ;  then 

(2)  ^=/(^)- 

Any  function  F(^x)  that  satisfies  this  equation  may  be 
symbolically  represented  by 

(3)  F(:x}=ffCx}dx; 

this  is  called  the  integral  of  f(x)' 

The  process  by  which  (3)  is  derived  from  (2)  is  called 
integration. 

According  to  the  above  notation,  we  understand  by 
\f(x)dx^  a  function  whose  derivative  is  f(x).  This  is 
simply  a  convention.  Instead  of  writing  "the  function 
whose  derivative  is  f(x)^'^  we  write  j  f(x)dx.  We  might 
just  as  well  introduce  the  notation  If(x)^  or  \f(x)\^  or 
any  other  which  we  might  choose.  /,  {]•>  \  ---dx^  would 
alike  be  symbols  meaning  "the  function  whose  derivative 


2.]     FUNDAMENTAL   CONCEPTIONS   OF  INTEGRATION     171 

is."  In  particular,  dx  is  a  part  of  the  symbol,  just  as  much 
as  the  second  brace  would  be  if  we  had  introduced  the 
symbol  \  \.  The  dx  has  no  meaning  in  itself,  but  taken 
with  the  I ,  it  means  "  the  function  whose  derivative  with 
respect  to  x  is."  It  might  seem  that  the  sign  I  would  alone 
be  a  sufficient  symbol,  and  ordinarily  it  would  be,  but  the  dx 
is  retained  as  part  of  the  symbol  from  historical  reasons, 
since  the  notation  is  firmly  imbedded  in  a  large  mass  of 
mathematical  literature,  and  also  because  it  is  often  neces- 
sary to  indicate  what  is  to  be  regarded  as  the  variable 
quantity  in  forming  the  derivative  in  question,  and  this  is 
conveniently  done  by  the  x  occurring  in  the  dx.  While  the 
symbol  I  -"  dx  has  the  meaning  just  indicated,  it  is  usually 
read^  as  we  have  already  mentioned  above,  "  the  integral  of  " 
whatever  function  may  occur  in  the  symbol,  and  if  it  be 
necessary  to  specify  what  is  to  be  regarded  as  the  variable, 
the  symbol  is  read  "  the  integral  with  respect  to  x  of." 

We  have  said  that  dx  is  to  be  regarded  as  a  part  of  the 
symbol,  yet  it  is  not  arbitrarily  fixed  so.  It  is  rather  the 
trace  of  a  finite  quantity  which  was  made  to  approach 
the  limit  zero  in  the  earlier  history  of  the  function  under 
consideration.  We  have  seen  this  illustrated  in  the  first 
paragraph  of  this  chapter,  and  shall  see  it  still  more  clearly 
when  we  come,  in  Chapter  VIII.,  to  a  second  definition  of  an 
integral  which  is  there  developed. 

On  applying  this  notation  to  the  examples  considered  on 
pp.  166-169,  we  obtain 

l=fgtdt  =  ^gt\ 

and  t=  Cl-l_dx=\\og-^. 

^  k  a  —  X  k       a  —  X 


172  CALCULUS  [Ch.  V. 

In  the  Differential  Calculus  we  grew  accustomed  not  to 

keep  — -,  the   symbol  for  the  operation   of   differentiation, 
ax 

invariably  separated  sharply  from  the  quantity  to  which  the 
operation  was  to  be  applied,  but  sometimes  to  write  for  com- 
pactness -^,     ^    7~^    ^  ^^^">  instead  of  -— y,  -r-(l  +^^),  etc. 
ax  ax  ax      ax 

The  notation  for  integrals  is  often  made  a  little  more  com- 
pact in  a  similar  manner.     Thus,  instead  of   i  -  dx,    I  1  dx, 

Jd       r  ^  X        ^ 

— ,    (  dx,  etc.,  unity  being  omitted 

in  the  latter  in  accordance  with  the  general  custom  of 
omitting  unity  when  no  confusion  is  caused   by  doing  so. 

In  every  case  I  "•  dx  constitutes  the  symbol  of  integration, 
and  the  remainder  of  what  is  written,  the  function  to  be 
integrated. 

Art.  3.  The    integral   calculus   as   an    inverse    problem. 

The  question  now  arises :  How  can  the  integral  F{x^  of  a 
given  function  f(x)  be  determined  ?  Before  taking  up  this 
question  we  note  that  the  operations  of  the  Differential  and 
the  Integral  Calculus  are  opposite  in  character. 

The  contrast  between  the  Integral  and  the  Differential 
Calculus  is  analogous  to  that  between  multiplication  and 
division,  or  between  involution  and  evolution ;  and  in  each 
of  tliese  cases  the  inverse  operation  is  essentially  more  diffi- 
cult than  the  direct.  While  we  secured  in  the  Differential 
Calculus  a  general  method  for  finding  the  derivative  when 
the  function  is  given,  there  exists  no  corresponding  general 
method  for  the  inverse  problem;  each  case  requires  special 
treatment. 

From   what   has   been   stated    above,   F(x)    or    \f(x)dx 


2-3.]   FUNDAMENTAL   CONCEPTIONS   OF  INTEGRATION  173 

is  to  be  understood  as  a  function  whose  derivative  is  f{x}^ 
as  shown  in  equation  (2),  p.  170.  If  we  substitute  in  this 
equation  for  F{x}  its  value  given  in  equation  (3),  p.  170, 
we  have 

(1)  j-Jf(x)dx=f(x), 

an  equation  which  proves  that  the  operations  of  differentia- 
tion denoted  by  — ■  and  of  integration  denoted  by  J  •"  dx 

counteract  each  other,  just  as  is  the  case  with  the  operations 
of  extracting  roots  and  raising  to  powers. 
In  a  similar  manner  we  obtain 

(2)  J[£#(x)]rfx-  =  J'(^), 

proving  again  that  —  and  (  •"  dx  have  contrary  effects. 
(XX  *^ 

If,  for  brevity,  we  put  u  —  F(^x),  equation  (2L)  takes  the 
simple  form 

(3)  (^^  (lx  =  u, 

J  doc 

an  equation  which  we  shall  have  occasion  to  use  repeatedly. 
Equations  (1)  and  (2)  show  that  the  derivative  of  an 
integral  as  well  as  the  integral  of  a  derivative  always  gives 
the  original  function,  just  as  the  nth.  root  of  an  n\h.  power,  or 
the  nth  power  of  an  nth  root,  always  gives  the  fundamental 
number.  For  every  two  inverse  kinds  of  calculation  there 
exist  two  equations  of  this  sort.  If  we  take  the  raising  to 
powers  as  corresponding  to  differentiation,  and  the  extrac- 
tion of  roots  as  corresponding  to  integration,  the  analogue 
of  the  first  equation  is 


174  CALCULUS  [Ch.  V. 

where  the  root  of  a  is  first  taken  and  the  result  is  raised  to 
a  power.     The  analogue  of  the  second  equation  is 

where  a  is  first  raised  to  a  power  and  the  root  of  the  result 
taken.  Likewise  in  (1)  above,  f(x)  is  first  integrated  and 
the  result  differentiated,  while  in  (2),  F(x)  is  first  differ- 
entiated and  then  the  result  integrated. 

Akt.  4.  The  constant  of  integration.  If  F{x)  be  an 
integral  oi  f(x)  so  that  it  satisfies  tlie  equation 

it  is  clear  that 

(1)  Fix')+0, 

where  (7  denotes  a  constant  quantity,  also  has  this  property; 

for  (p.  128) 

d[F(x)±C^_dF(x)_ 

dx  ~     dx     ~'^^^^- 

The  function  F(x)  -f-  C  can  therefore  be  also  regarded  as 
an  integral  of  /(a;) ;  in  other  words,  a  given  function  f(x) 
has  a  boundless  number  of  integrals  ;  every  value  of  C  deter- 
mines one  of  them. 

This  may  be  expressed  by  the  formula 

(2)  ff(x)  dx  =  F(x)  +  C.      . 

There  is  no  danger,  however,  that  we  shall  run  into  error 
by  usually  hereafter  writing  our  equations  as  we  have  thus 
far  done,  simply 

"^fix^x^FCx) 


p 


for   the   exact   reading   of    this   equation   is:    one   integral 
function  of  f(x)  is  F(x)^  while  that  of  equation  (2)  is:  all 


3-5.]  FUNDAMENTAL   CONCEPTIONS  OF  INTEGRATION  175 

functions  of  the  form  F(x^  4-  0  are  integral  functions  oif(x). 
The  constant  C  is  called  the  constant  of  integration. 

Art.  5.  The  fundamental  formulae  of  the  Integral  Cal- 
culus. Inasmuch  as  the  Integral  Calculus  is  the  inverse  of 
the  Differential  Calculus,  a  formula  of  the  Integral  Calculus 
may  be  derived  from  any  formula  of  the  Differential  Cal- 
culus.    Thus,  if  the  equation 

be  considered,  it  is  clear  that  the  equation 
jfix^dx  =  F(x) 

may  be  at  once  deduced  from  it. 

In  this  way  a  first  set  of  integral  formulae  may  be  obtained 
immediately  from  the  formulsB  established  in  Chap.  IV. 
Thus,  for  the  power  x^'^'^  we  have 

ax 


d'  ^"" 


or 


n-\-l 

'  =x" 


dx 

x'^dx  = -• 

Similarly,  the  formula  ' 

d(—GOSX^ 

— ^ — —  sm  X 

dx 

may  be  expressed  in  another  form  as 

I  sin  xdx=  —  cos  x. 
».' 

In  this  wise  we  obtain  the  following  preliminary 
13 


176  CALCULUS  ,  [Ch.  V. 


TABLE   OF  INTEGRALS 


1.  \afdx  = -•  5.     I    .   .^     =  —  cot  X. 

^  n  -{-1  ^  sm^  X 

2.  I  (i0^xdx  =  ^inx.  6.     i  e^dx=  e^. 

3.  I  sin  xdx=  —  cos  a;.  7.     I  a^c?a;  = 

*/  *^  log  a 

4.  C-^=tanx.  8.      r^=log2:.* 

*^  COS^  X  ^     X 


■/; 


6?:r 


=  arc  sin  x. 


Vl  -  x^ 

/dx 
=  arc  tan x=  —  arc  cot x.  f 
1  +ar 

Art.  6.  The  geometric  signification  of  the  constant  of 
integration.  There  is  but  little  in  elementary  mathematics 
analogous  to  the  boundless  number  of  integrals  belonging  to 
one  and  the  same  derivative,  f{x)>  Indeed,  in  extracting 
the  square  root  of  a  quantity  we  obtain  two  solutions,  and  a 
cube  root  has  three  values,  but  nowhere  does  there  occur  as 


*  The  integral  on  the  left  side  is  formally  contained  in  the  first  integral 
of  the  Table.  But  in  that  case  the  first  integral  formula  refuses  to  work, 
since  w  +  1  =  0 ;  the  above  formula  is  therefore  substituted  for  it.  This 
illustrates  how  the  Integral  Calculus  may  lead  us  to  new  functions.  If 
we  had  not  previously  defined  logarithms  and  studied  their  properties, 
in  particular,  their  derivatives,  we  should  not  be  able  to  find  the  integral 

Cdx 
denoted  by  J       ;  but  the  proposing  of  this  very  simple  function  for  integra- 
tion might  lead  us  to  study  the  function  whose  derivative  is  -,  and  thus  to 
discover  the  fundamental  properties  of  logarithms. 

t  This  result  means  that  arc  tan  x  and  —  arc  cot  x  are  both  functions  that 
have  as  derivative  ;   that  is,  they  are  both  integrals  of -,  and 

can  therefore  differ  only  by  a  constant;  this  we  know  from  trigonometry 
to  be  the  case. 


6-6.]  FUNDAMENTAL   CONCEPTIONS  OF  INTEGRATION  177 

here,  in  finding  the  integral  function  for  the  derivative  f(x)^ 
an  unlimited  number  of  results.  The  formal  explanation  of 
this  point  has  been  presented  previously.  We  now  proceed 
to  discuss  the  geometric  meaning  of  the  constant  of  integra- 
tion. 

Let  F(x)  be  any  integral  of  f(x)^  so  that 

dFCx)      ^r  \ 

If  y  =  Fix}, 

this  equation  represents  a  curve,  for  which  (p.  11^ 

this  equation  determines  the  position  of  the  tangent  at  any 
point  in  our  curve.  The  determination  of  the  integral  of 
the  given  function  /(x)  then,  geometrically  speaking, 
amounts  to  determining  the  ordinate  ^  of  a  curve  when  the 
direction  of  the  tangent  at  every  one  of  its  points  is  known. 
It  is  easily  seen  that  the  problem  of  constructing  a  curve 
from  a  knowledge  of  its  tangents  leads  to  a  countless  number 
of  curves.  For  we  observe  that  the  determination  of  the 
tangents  by  means  of  equation  (1)  is  made  so  that  the  angle 
T  in  any  point  of  the  curve  depends  only  upon  the  abscissa  x 
of  that  point.  If  any  curve  whatever  be  known  that  fulfils 
the  conditions  of  the  problem,  and  it  be  moved  any  given 
distance,  parallel  to  itself,  in  the  direction  of  the  axis  of  y,  it 
will  in  its  new  position  also  satisfy  the  conditions  of  the 
problem.  For  example,  if  A^  (Fig.  46)  be  a  point  which 
reaches  the  position  B^^  both  A^  and  B^  have  the  same 
abscissa,  and  their  tangents  have  remained  parallel  through- 
out the  motion. 


178 


CALCULUS 


[Ch.  V. 


We  can  show  the  same  thing  in  another  manner,  which 
also  furnishes  us  with  a  means  of  actually  constructing  the 
curves  under  discussion  when  equation  (1)  is  given,  and  thus 

of  determining  the  required  in- 
tegral graphically.  Let  A^ 
(Fig.  46)  be  any  point  of  the 
plane,  and  let  its  coordinates  be 
^v  Vv  ^^  determine  an  angle 
Tj  by  means  of  the  equation 


tan  Ti  =/(a^i), 


Fig.  46. 


and  draw  a  straight  line  through 
A^^  making  this  angle  with  the 
cc-axis.  On  this  straight  line  we  take,  near  to  J.^,  another 
point  A^  whose  abscissa  is  a^g,*  and  calculate  another  angle 
Tg  from  the  equation 

tanT2=/(a:2). 

We  now  draw  through  A^  a  straight  line  with  slope  Tg,  and 
take  upon  it  a  point  A^  near  to  A^ ;  the  abscissa  of  this 
point  being  a^g,  the  angle  Tg  is  determined  by  the  equation 

tanTg=/(a;3). 

By  continuing  thus,  we  get  a  series  of  lines  through  A^^  A^^ 
^g,  •••,  which  form  with  the  a;-axis  the  same  angles  as  do 
the  tangents  of  the  required  curve.  The  nearer  the  points 
J.J,  J.2,  ^g,  •••,  are  together,  the  closer  the  polygon  approaches 
the  curve,  and  its  limit  is  the  curve  itself.      In  this  way 


*  It  is  simplest  to  choose  the  abscissae  xi,  iC2,  ^z-,  •••,  so  that 

X2  —  X\  —  Xz  —  X2  —  '"  '■, 

for  example,  Xi  =  5,  X2  =  5.1,   xz  —  5.2,  etc. 


6.]     FUNlJAMtJNTAL   CO]StCi:PTlONS  OF  INTEGRATION     179 

we  have  actually  constructed  a  curve  corresponding  to  equa- 
tion (1). 

Since  we  had  perfect  freedom  in  the  choice  of  the  starting 
point  A^  for  our  construction,  the  value  of  the  ordinate  i/^ 
belonging  to  the  abscissa  x^  is  arbitrary  also.  If  we  substi- 
tute for  this  value  of  i/^  another  value,  as  Fj,  corresponding 
to  the  point  B^^  we  can  in  a  similar  manner  draw  the  sides  of 
a  polygon  B^^  B^^  B^^  •••,  belonging  to  another  curve  also 
representing  an  integral.  Since  only  the  abscissae  x^,  x^^ 
a^g,  •••  occur  in  the  equations  for  r^,  r^,  Tg,  •••,  we  see  imme- 
diately that  the  sides  B^B^,  ^2^3^  ^3^4'  "*  ^^^^  parallel  to 
the  sides  A^A^,  ^2^3'  ^s^v  '"'     Therefore, 

(2)  A,B,=  A,B,=  A,B,=  >>^, 

or  ^1  -  ^1  =  ^2  -  ^2  =  ^3  -  ^3  =  -  ; 

if  we  denote  by  0  the  constant  value  of  this  difference,  we 
obtain  linally 

Fi  =  ^1  +  (7,  r2  =  ^2  +  ^^  ^3  =  ^3  +  ^- ; 

that  is, 

(3)  r=2/  +  C. 

If  the  equation  of  the  first  curve  be  written  in  the  form, 

(4)  y  =  F(x-), 

then  the  equation  of  the  second  curve  becomes 

(5)  Y=F(x)+0, 

and  this  is  the  equation  we  started  out  to  explain.* 


*  The  above  considerations  show  further  that  every  integral  function  is 
deduced  from  F(x)  by  the  addition  of  a  constant ;  on  p.  174  we  have  proved 
only  that  all  functions  of  the  form  F(x)+  C  are  integral  functions. 


180  .     CALCULUS  [Ch.  V. 

We  draw  the  conclusion,  further,  that  in  fixing  upon  any 
particular  integral  among  the  total  number  of  possible  ones, 
we  are  perfectly  free  to  prescribe  the  value  which  the  func- 
tion shall  have  corresponding  to  any  one  given  value  of  x. 

Abt.  7.  The  physical  signification  of  the  constant  of 
integration.  To  illustrate  the  physical  signification  of  the 
constant  (7,  and  to  understand  the  necessity  of  its  occurrence, 
we  begin  with  the  motion  of  freely  falling  bodies,  for  which 

whence 

(2)  V  =^gdt, 

or,    (3)  v  =  gt  +  0. 

We  have  already  considered  the  motion  of  freely  falling 
bodies  (p.  167),  and  found  there  that  v  =  gt.  But  we 
treated  then  only  motion  in  which  the  falling  body  was 
at  rest  at  the  beginning  of  its  fall,  that  is,  at  the  moment 
^  =  0.  The  laws  of  freely  falling  bodies  embrace  also  the 
cases  of  motion  in  which,  at  the  instant  when  gravitation 
commences  to  act,  the  body  in  question  already  possesses 
a  certain  velocity  of  its  owu,  which  may  be  directed  up- 
wards as  well  as  downwards. 

In  order  to  single  out  any  one  of  these  motions,  we  must 
know  what  its  velocity  V  is  at  some  moment.  It  is  advis- 
able to  take  the  moment  when  ^  =  0,  since  the  initial 
velocity  is  usually  given.  Let  this  initial  velocity  be  Vq,  so 
that  substituting  i  =  0  and  v  =  v^^  in  (3),  we  see  that 

(4)  ■  v,=  0; 


I 


6-7.]  FUNDAMENTAL   CONCEPTIONS   OF  INTEGRATION  181 

and  for  the  motion  in  question  we  have  then 

(5)  v  =  gt-{-  Vq, 

which  determines  the  velocity  v  at  any  given  moment  t. 

If  the  body  be  thrown  vertically  upward,  its  initial  velocity 
is  negative  ;  we  put 

and  have  v  =  gt  —  V, 

The  question  may  arise :  when  does  the  body  come  to  rest, 
or  change  the  sense  of  its  motion  ?  This  occurs  when  v 
becomes  equal  to  zero;  the  corresponding  value  of  t  is 
derived  from  the  equation 

0  =  gt-V, 
and  is 

V 

(6)  t  =  ~. 

9 

We  next  take  up  the  discussion  of  the  inversion  of  sugar, 
for  which  we  have  already  (p.  169)  established  the  equation 

(7)  ^^  1 


dx      k(a  —  x) 

which  by  integration  (with  the  addition  of  the  constant  (7) 
becomes 

(8)  «  =  hog^— +  C., 

k        a  —  X 

C  must  naturally  have  a  definite  value  for  any  given  reac- 
tion, and  this  value  can  be  determined  as  follows.  If,  as  is 
customary,  the  time  be  counted  from  the  moment  when  the 
reaction  begins,  the  mass  of  sugar  inverted  at  the  time  ^  =  0 
is  a;  =  0,  and  we  have  the  equation 


182  CALCULUS  [Ch.  V. 

which  fixes  the  value  of  O.     If  we  substitute  this  value  of 
0  in  equation  (8),  we  find 

1        .    * 

=  7  log 


k        a  —  X 

In  practice,  the  constant  0  is  generally  determined  in 
another  way.  A  direct  observation  is  made  of  the  mass  of 
sugar  a^i  inverted  in  a  time  t^.     Then  we  have  the  equation 

from  which  the  value  of  0  may  be  found.     If  this  value  of  C 
be  substituted  in  equation  (8),  it  follows  that 

t^-t=-  log log 


k        a  —  x^      k        a  —  X 

=    log *, 

k       a  —  x^ 

and  finally 

(11)-  k  =  -^\og^^^. 

t-^  —  t       a  —  x^ 

This  is  the  best  form  that  our  equation  can  be  made  to 
assume  for  its  experimental  corroboration.  It  sho>vs  that 
its  right-hand  member  must  be  a  constant,  and  it  is  easy  to 
find  out  whether  or  not  this  is  the  case  by  substituting 
various  values  of  ^,  with  the  corresponding  values  of  a:,  f  as 
found  by  experiments. 

*  Formula  6,  Appendix. 

t  See  the  applications  in  Chapter  VII.,  pp.  227,  244. 


7.]    FUNDAMENTAL   CONCEPTIONS  OF  INTEGBATION     l88 


Exercise.  Long  (Journal  of  the  American  Chemical  Society,  Vol. 
XVIII,  p.  129,  1896)  found  the  following  amounts  of  sugar  x  inverted 
in  the  times  t : 


a  =  43.91 

«  =      30                 60                 120 

180 

300 

x=    3.91              7.56              14.61 

19.06 

28.09 

Compute  the  values  of     ^     log  ^  ~  ^  .* 

*The  formula  is  true  for  Naperian  logarithms  only,  but  common  loga- 
rithms may  be  used  here,  since  we  wish  simply  to  verify  that  the  value  is 
constant. 


CHAPTER   VI 

THE  SIMPLER  METHODS  OF  INTEGRATION 

Art.   1.  Integration  of  sums  and  differences.     In  Chap- 
ter III,  p.  127,  we  saw  that 

dx  dx      dx 

and  by  integrating  this,  we  obtain 

•^  \dx      dxj 

But  (p.  173),   u=  \  -^  dx,  and  v=  |  —-dx; 
^  dx  ^  dx 

hence 

in  words,  the  integral  of  a  sum  of  two  terms  is  equal  to  the 
sum  of  the  integrals  of  the  separate  terms. 

A  similar  formula  evidently  holds  for  the  sum  of   any 
given  number  of  terms. 

Likewise,  by  integrating  the  expression 

/Q\  ^  r    —    's_du_dv 

dx  dx      dx 

we  find  that 

^  ^  J  \dac     doc  J  J  djp  J  da) 

in  words,  the  integral  of  a  difference  is  equal  to  the  difference 
of  the  integrals. 

184 


1.]  THE  SIMPLER  METHODS   OF  INTEGRATION  185 

Finally,  we  obtain  by  integrating  the  formula 

y'r''\  Ct       y'  ^  CtU 


in  which  a  denotes  a  constant, 


^     ax 


u=  f  —~dx, 


'       ' 


du 
au=  t  a 

But 

u=  I 

dx 

and,  accordingly, 

(6)  i  a—-  dx  =  a  (  — -  dx,       *      .  " 

^     dx  ^  dx  «       • 

This  formula  shows  that  any  constant  facjbor  of  a  function 
given  for  integration  may  be  written  before  or  after  the  sign 
of  integration.  < 

EXAMPLES 

1.  ^Uh^.,h'^  -c'I^]dx  =  ai'^dx  +  h  ('^dx  -  c  (^^dx. 

J  \    dx        dx        dx  I  'J  dx  J  dx  J  dx 

2.  \  (x  +  sin  x)dx  =  \  xdx  +  \  sin  xdx  = cos  x  +  C. 

3.  I  (ax^  +  bx  4-  c)dx  =  i  ax^dx  +  I  bxdx  +  \  cdx 

N      '  =^+ht-^cx-i-  C. 

3  2 

4.  ^(ax^  +  -\dx  =  ^ax^dx+^^-^dx  =  ^+b\ogeX+C, 

5.  I  (rt  cos  X  +  b  sin  x)dx  =  a  sin  x  —  b  cos  x  +  C. 

EXERCISES   XIX 

Note,  In' these  exercises,  and  in  all  integrations,  it  is  always  a  suffi- 
cient proof  of  the  correctness  of  a  result,  to  show  (by  diiferentiation) 
that  the  derivative  of  the  result  obtained  is  the  function  given  for 
integration. 

Write  the  integrals  of  the  following  functions  : 

1.  xK  3.    3x.  5.    ax^.  7.    -ex. 

2.  x^.  4.   x-^  6.    a:*  -  d.  8.    3  +  x. 


1S6  CALcnm  [Ch.  vt 

J.9. 


11.   x3  4.  3a;2+4a:-6.  ""-4  **"  4^^ 

Write  the  result  of  the  operations  indicated  in  the  following  expres- 
sions : 

16.  fs  cos xdx.  22 .    ( I- -]- 7^  dx. 

17.  ^(^x  +  ^yjx.  23.   j- 5 rf:,. 


18 
19 

20.     f-^^:..  26.    f^. 

(7  cos  a:  -  4  sin  x  +  1)  dx.  27.   J  - 

28.    f  (e^  +  a''  +  x")  dx. 


2\/l  -a;2 


•     1  ( i-  +  -n~  r^'  25.    CoCsin  X  +  cos  a:)  ria;. 

»/  Vcos^a;      sin^a:/  J    ^  ^ 


dx 
^~x 


Art.  2.  Integration  by  the  introduction  of  new  variables. 
The  determination  of  the  integral  can  often  be  facilitated  by 
the  introduction  of  new  variables,  analogously  to  the  pro- 
cess which  has  already  been  explained  for  the  Differential 
Calculus  (pp.  152-153). 

Let  there  be  given  for  integration 

J  f(x)dx, 

and  suppose  that  the  desired  integral  is  <^(a^),  so  that  we 
have 

(1)  <l>ix)=ffCx}dx, 

(2)  or  /(^)=£c^(:,). 

Now  put 

(3)  x  =  ylr(iu}. 


I 


1-2.]  THE  SIMPLER  METHODS   OF  INTEGRATION  187 

Then  f(x)  as  well  as  </)(a;)  are  functions  of  u^  and  we  have 

(p.  152) 

d(f>  (x)  _  dcf)  (po)      dx  ^ 
du  dx         du 

or,  making  use  of  (2), 

du  du 

Whence,  integrating  with  respect  to  u^ 

(4)  <i>(x)=^f{x)^du. 
Equating  values  of  (i)(x)  from  (1)  and  (4),  we  have 

(5)  ^f{ic)d^=^f{x)^du, 

where,  of  course,  at  some  convenient  point  in  the  simplifica- 
tion of  the  expression  under  the  integral  sign  in  the  right 
member,  x  must  be  replaced  by  its  value  in  terms  of  u  from 
(3),  so  that  the  function  to  be  integrated  with  respect  to 
u  is  finally  expressed  in  terms  of  u  alone. 

We  now  apply  this  method  to  several  examples. 

I.    Given  \(a-\-xydx. 

Put  a  +-2:  =zu^ 

dx  _^ 

du 

Hence  j  (^  +  ^ydx  =  j  u"-  •  -^du=  j  u"du, 


and 


I  u^du  = 


n  +  1 


188  CALCULUS  [Ch.  VI. 

Restoring  the  values  of  w, 

(6)  f(a  +  xydx  =  ^^  +  ^]"^'  +  C* 

*/   ^  n  -\-  1 

In  particular,  we  have  for  various  values  of  n, 

r — dx^  C(a-{-x)-^dx  =  — ^+  a 

J  (^a-{-xy     J^  ^  a  +  x 

J  {a  +  xy     J^  "^  2{a-i-xy        ' 

etc.,  etc. 

II.    The  integral 

I  (a  —  xydx 

may  be  treated  analogously. 
We  put  a  —  X  =  u^ 

whence  ^  =  -  t, 

J"(«  -  ^)"(^:r  =X^"  •  (-  1)  •  (^^^  =  -ye'du, 

(a  —  :r)"cZ2;  = ; 

n  -\-l 

and,  replacing  u  by  its  value, 


*  It  is  most  convenient  not  to  add  the  constant,  until  the  final  form  of  thu 
result  is  reached. 


2.]  THE  SIMPLER  METHODS   OF  INTEGRATION  189 

and  in  particular, 

f(a  -  x)dx  ^  _  (^  -  ^)%  0^ 

{{a  -  xydx  =  -  ^^  7  ^^^  +  C, 

f-^-^^o  =  fC^  -  ^)"^^^'^  = ^ +  <^. 

^  (a-xy     ^  ^  ^  2(a  -xy 

etc.,  etc. 

If    w  =  —  1,   the   integrals   in    both   these   cases   lead   to 
logarithms  (cf.  p.   176). 

For  example,  if  in  the  integral 

Jdx 
a  -\-  X 

we  put  a  +  X  =  u^ 

and  therefore,  —  =  1, 

du 

we  have  ' 

(8)  r-^=r*^  =  logM  =  log(a  +  x)+a 

^  a  +  X     ^    u 

III.    To  determine  the  integral 

rAdx 
^  a  —  X 
we  put  a  —  X  =  u. 

Therefore,  —=-1, 

du 

and  r-^^  =  -^r^^-^log^  =  ^logl, 

(9)  *  =^log— 1_  +  (7. 

a  —  r^; 


190  CALCULUS  [Ch.  VI. 

We  have  already  met  the  last  integral  in  the  consideration 
of  the  inversion  of  sugar  (p.  182).  There,  however,  we 
simply  verified  the  result  which  we  have  here  deduced. 

IV.    Given  j  tan  x  dx. 

We  write,  in  the  first  place, 

/tan  xdx=  \  dx^ 
*^  cos  a: 

and  put  cos  x  =  u^  or  x  =  arc  cos  u^ 

whence 

dx  1  1  1     ^ 


du  Vl-i*^  VI- cos2:c  sin  a; 

We  have  then        |  — —  dx=  —  I  —  =  —  losr  u. 
J  cos  a;  J    u  ^ 

Consequently,! 

(10)  I  tan  xdx  =  log =  —  log  cos  x  +  0. 

^  cos  X 

V.    Similarly,  in    ^ o^ot x dx  =\^-^ dx, 
^  ^  sin  X 

we  put  sin  x^u, 

whence  ^  =  _1_, 

du     cos  X 
and  find 

(11)  1  cot  xdx  =  log  sin  x  -\-  C. 

*  This  can  be  found  also  thus : 

—  =  -sm  X,  and  (p.  142)  v-  =  :t-  = 


dx  '^  du     du         sin  x  y/i  _  y;2 

dx 
t  Formula  29,  Appendix. 


2-3.]  THE  SIMPLER   METHODS   OF  INTEGRATION  191 

VI.    Given  1  sin  x  cos  x  dx. 

Put  sin  x=  u^  whence     -—  = ,  and  therefore, 

du     cos  X 

(12)  \  ^u\xQo^xdx=  \  udu= —  =  — \-Q, 

EXERCISES  XX 

Determine  the  following  integrals : 

1.  ye^dx.                                             10.  xx^co^xMx            (put  a:  =  m^). 

2.  p^  '^    dx.                                      11.  J  6^008  6==                 (putc»=  =  M). 

3^    r oMx ^2.    r^x^-icosx"  (putx'»  =  M). 

J(a:5_73)73  J  ^^  >' 

4.  ^{x''-bx){2x-b)dx,  13.  J^HLdM^)!^    (putlogx=:w). 

5.  (-l£^dx.  14.    ri±-?2^rf^. 
»^  Va^  —  x'i  •^  :r  +  sin  x 

6.  ^(Sx^  +  5x-iy(Qx  +  5)dx\      15.   J — ^-^  (put  1  -  a;  =  m). 

7.  f(2x3-7a:2+3)»(6a:2-14a:W  16.    f-^^. 
•^  •^  8111  ma: 

8.  (      ^^      (put  w  -  a:  =  Va^  +  x^).  17.    j  sin^  a;  cos^  x  dx  (put  cos  a:  =  m)  . 

9.  f22!^c/a;  (put  a:  =  m^).  iS.     (Va^-  x^dx  (put  a;  =  a  sin  m). 

•^     y/x  ^ 

Art.  3.  Integration  by  parts.      By  integrating  the  for- 
mula for  the  differentiation  of  a  product,  viz.  (p.  130) 

^1^  d  .     ^        du  ,      dv 

(1)  ^(uv}=v—  +  u~, 

ax  dx         dx 

we  find  uv  =  I  v-—-dx+  )  u^r  ^^' 

^    dx  ^     dx 

U 


192  CALCULUS  [Ch.  VI. 

the  inverse  of  the  analogous  formula  of  the  Differential 
Calculus.     By  writing  this  formula  in  the  form 

(2)  (u^d3c  =  uv-iv^  die, 

J     dx  J     doc 

we  see  at  once  that  it  expresses  one  integral  in  terms  of 
another.  If  we  know  the  integral  in  the  right  member, 
we  can  by  its  aid  calculate  the  integral  of  the  left  member 
also.     The  method  of  applying  this  formula  is  to  determine 

functions  2^  and  v  such  that  the  product  u—  is  equal  to 

dx 

the  function  given  for  integration,  and  therefore  an  equation 
of  the  type  (2)  can  be  set  up  having  our  desired  integral  as 
its  left  member.  How  this  is  to  be  done  in  practice  will 
appear  best  from  the  consideration  of  some  examples. 

I.    In  order  to  apply  our  formula  to  the  integral 


j  log  X  dx, 


u  = 

=  log  X, 

dv 
dx 

=  1, 

du 
dx 

_1 

X 

v  = 

■-X, 

we  put 

and  then  find 

whence  j  log xdx  =  x log x—  \  X'-dx 

(3)  =  X  log  x  —  x. 

II.    As  a  second  example,  we  take  the  integral 

j  xe^  dx. 
In  this  case  we  put 


dv       X 
u  =  x,  — -  =  e  , 

dx 

and  find  -—  =  1,  v  =  \  e^  dx  =  e^, 

dx  ^ 


3.]  THE  SIMPLER  METHODS   OF  INTEGRATION  193 

Therefore  it  follows  that  (substituting  in  (2)), 

j  xe^  dx  =  xe^  —  \  e^  dx 

(4)  =  xe^  —  e'^. 

No  general  rule  can  be  given  as  to  the  manner  in  which 

an  integral  is  to  be  divided  into  the  two  parts  u  and  —  in 

dx 

order  that  the  method  may  actually  accomplish  its  object ; 
this  can  be  ascertained  only  by  trial.  It  is  sometimes  advan- 
tageous to  take  unity  as  one  of  the  factors ;  for  example, 
\\ogxdx  was  found  thus.  It  is,  however,  clear  that  the 
function  v  must  in  every  case  be  so  chosen  that  it  is  possible 

to  determine  it  from  its  derivative  — ,  and  that  the  integral 

dx 

to  which  the  required  integral  is  reduced  must  be  known  or, 

at  least,  must  be  more  easily  determinable  than  the  required 

one. 

III.    If,  for  example,  we  put  in  the  last  integral, 


dv 


u  ==  g-*,  —-  =  x^  I 


dx 

which  is  perfectly  permissible,  then  v  is  in  this  case  also 
easily  determined,  for 


v=^fx 


X  diji 

dx  =  — ,  and  further,  —  =  e^^ 

2  dx 


whence  \xe^dx=  ^^^—  )  —e^dx. 

But  in  this  way  we  have  referred  the  required  integral  back 
to  one  that  is  evidently  more  complex  ;  such  a  substitution 
has  therefore  no  practical  value  for  the  determination  of  the 
integral  sought. 


194  CALCULUS  [Ch.  VL 

The  method  of   determining  integrals   just   described  is 
called  the  method  of  integration  by  parts. 

IV.    We   next  employ   this   method   in   determining  the 
integral, 

j  X  sin  X  dx. 

By  putting  u  =  x^  —  =  sin  ic, 

(XX 


we  find 


du 


dx 
and  therefore 


=  1,  t;  =  J  sin  xdx=  —  cos  x^ 


1  X sin xdx=  —X cos x  -{-  \  cos x dx 

(5)  =  —  X  cos  X  +  sin  a;. 


V.    The  integral  i  x^  sin  a: 


c?a; 


can  be  treated  in  a  similar  manner.     We  put 

9    dv 
u  =  x^f   —  =  sm  X, 
dx 

Then,  -^  =  2  a:,    v  =  \  sin  xdx=  —  cos  a; ; 

dx  ^ 

whence         j  a^^  gin  xdx=  —  x^  cos  x -\-  \  2  x  cos  a;  c?a;. 

In  order  to  determine  the  integral  in  the  right-hand  mem- 
ber, we  put 

o       dv 
u  =  2x^   -z-=  cos  X ; 
dx 

whence  -—  =  2,         v  =  j  cos  xdx  =  sin  a;, 


3-4.]  THE  SIMPLER  METHODS  OF  INTEGRATION  195 

and  find  accordingly,  '     • 

j  2  a;  cos  xdx=2xsmx—  j  2sinxdx 
=  2x sin x-\-2  cos x ; 
obtaining  as  the  final  result 

(6)  j  x^  sin  X  dx  =  —  x^  cos  a:  +  2  a;  sin  x  -\-2  cos  a:. 

EXERCISES   XXI 

Integrate  by  parts : 

1.  \  X  cos  nx  dx.  5.     |  arc  sin  a:  rfa:. 

2.  I  a;2  sin  na;  dx.  6,     Ca:^^"*  f/a:. 

3.  ^x^e^dx.  ^    ^x^SLYGsinxdx.      • 

4.  i  a:^  log  X  c?a:. 

Show  that : 

8.     I  xe^'^'dx  =  — ^^ — ^• 

J  ^  (n  +  1)2 

10.    j  arc  cot  xdx  =  x  arc  cot  a:  +  |  log  (1  +  x^). 

Art.  4.  On  special  artifices.  The  examples  hitherto 
treated  are  already  sufficient  to  show  that  the  evaluation  of 
integrals  is  markedly  more  complicated  than  the  formation 
of  derivatives.  This  corresponds  to  the  character  of  the 
Integral  Calculus  as  dealing  with  an  inverse  problem.  In 
particular,  we  do  not  have  in  the  Integral  Calculus  a  defi- 
nite method  corresponding  to  that  for  forming  derivatives, 
enabling  us  to  form  the  integrals  of   arbitrary  functions. 


196  CALCULUS  [Ch.  VI. 

Consequently,  the  determination  of  integrals  makes  drafts 
on  our  resources  quite  different  from  those  made  in  the 
formation  of  derivatives.  Mathematical  science  is  as  yet  far 
from  able  to  determine  the  integrals  of  arbitrarily  assigned 
functions,  and  it  does  not  even  fall  within  the  scope  of  this 
work  to  treat  or  to  enumerate  all  the  results  which  have 
thus  far  been  obtained  for  integrals  of  special  form,  more 
or  less  complicated.  We  add,  however,  several  simple 
examples  illustrating  some  of  the  artifices  by  which  many 
integrals  may  be  determined. 

Art.  5.  Integration  by  transformation  of  the  function  to 
be  integrated.  It  frequently  happens  that  the  function 
to  be  integrated  may  be  transformed  so  as  to  bring  the 
integral  under  results  previously  found. 

I.  To  determine 

(1)  f.  '^^ 

^  sin  X  cos  X 
We  have 

/dx        _  C'^iv^x -\- ao^'^Xn   ^ 
sin  X  cos  X     ^      sin  x  cos  x 

(2)  =C^J^dx+('^^dx 

^  COS  X  ^  sm  X 

and  accordingly,  introducing  the  results  of  p.  190, 

C       dx  1        .  1  1      sin  a; 

I  — =  log  sin  X  —  log  cos  X  —  log 

•'^  sm  X  cos  X  cos  x 

(3)  =  log  tan  x, 

II.  This  result  will  enable  us  to  compute  the  integrals, 

(4)  f-^  and    f-i^- 

^  sin  X  ^  cos  X 

*  Formula  28,  Appendix. 


4-6.]         THE  SIMPLER  METHODS   OF  INTEGRATION  197 

We  have 

dx        r         dx 


(5)  f4^=f- 


9X  X  ^ 

-  sill  -  COS  -  * 


O  i> 


Put  -  =  u->  and  hence  -— -  =  2. 

2  du 

(6)        f-i^^  r_^_^ :=  log  tan  2^  + (7=  log  tan  1  +  0. 

^  sm  a;     ^  sm  u  cos  ^^  2 

/'  dx 
-> 
cos  x 

we  first  have  f 

dx, 


sm  ( "I  +.  ^ 


(7)  .  f-^=C- 

^  cos  a;     ^ 

Now  we  put 

TT  ,  1  dx      -, 

--\-  x  =  u,  whence  -—  =  1 ; 
2  atfc 

and,  therefore, 

g  r_^^  r-^  =  logtan^=logtanr^  +  ^) 

^^       -'cosa^     -'sini^         ^        2^        ^        \4      27 

III.    Tofind  J'Va2-2;2c?:z;. 
Integrating  by  parts, 

(9)  fVa^  -  x^  dx  =  xVa'^  -  x^  +  f- 


— -^  C?2J. 


Va^  —  x^ 


(10)     r  ^^    d.  =  p^-^''^-^)  c^. 


»2 


«=  I  —  '          dx  —  I  -Va^  —  0?  dx 
=  a^  arc  sin |  Va'^  —  a:'^  c?a;. 


*  Formula  38,  Appendix.  -t  Formula  17,  Appendix. 


19B  CALCUtm  [Ch.VI. 

Substituting  in  (9), 

(11)  j  Va2_  x^  dx  =  x^a?  —  x^  +  a?  arc  sin j  V^^  —  a^^  c?^;. 

Whence 

(12)  I  Va^  —  a:2  c?^  = 1-  —  arc  sin  — 

IV.    We   could   find    f       ^         cZa:,  by  subtracting  (12) 
•^  Va2  _  x^ 

from  (9)  above.     We  may  also  find  it  as  follows : 

Integrating  by  parts, 

I     ,  dx=  —  xs/c^  —  ^  4-  I  Va^  —  x^dx 

^   a///.2  _  ^2  ^ 


=  —  rrVa^  —  a;2  +  I  -^ — — ^  dx 


^  ^a^-aP-     ^  Va2  _  x^ 
=  —  x^a?  —  xP  -\-  a?'  arc  sin I  - 


x^ 

dx. 


Whence 

3)  I  —  dx  =  —  -  Va'^  —  a;^  +  —  arc  sin  -• 

-^  Va2  _  :t;2  2  2  a 


V.    To  find  J'V^M^tZa:. 
Integrating  by  parts, 
(14)        J"v^2-:jr^2  ^^  ^  x^/W+^^  -  J"- 


a:2 


c?a;. 


V  a^  +  x^ 
We  have  also, 


Va2  4.  ^  •^^   Va2  +  :i:2      c/  y^2  _|.  ^2 


c?a:. 


5-6.]  TBE  SIMPLER  METHODS  OF  INTEGRATION  199 

Adding,  and  dividing  by  2, 

(15)      (^W^^^dx  =  ^  Va2  4.  a;2  +  ^  r — jg — 
-^  2  2*>'Va2  +  ^ 

=  -  Va^  +  x^  +  ■—  log  (a:  +  Va*-^  +  x^^. 

(See  Ex.  8,  p.  191.) 

Art.  6.  Formulae  of  reduction.  In  many  cases,  the 
given  integral  may  be  expressed  in  terms  of  known  func- 
tions and  a  new  integral.  The  determination  of  the  given 
integral  is  thus  reduced  to  the  determination  of  the  new 
integral,  and  the  formula  connecting  the  two  is  called  a 
formula  of  reduction.  The  method  of  integration  by  parts 
is  an  illustration. 

I.  Integration  by  parts  will  also  lead  to  many  special 
formulae  of  reduction.  Thus,  first  integrating  by  parts,  and 
then  in  the  resulting  integral,  multiplying  numerator  and 
denominator  by  Va^  —  x^^  and  simplifying,  we  find 

(1)  f        -^  J,^_^V^23^2^(^-1>Y"""^"     >^ 

By  applying  this  formula  repeatedly,  we  should  at  last 

/dx  C       X 

'  or  to   I  — n^^m  dx  (according 

as  n  is  even  or  odd),  both  of  which  are  known. 

II.  We  have 

(2)  —  tan  X  sec"~^  a:  =  (n  —  2)  tan^  x  sec"~^  x  +  sec"  x 
dx 

=  (ii  —  1)  sec"  ;r  —  (n  —  2)  sec"~^a; ; 

*  Equation  (1)  does  not  hold  for  n  =  0,  since  for  this  value  of  w,  the 
coefficients  become  infinite.  It  is  usually  possible  to  see  readily  for  what 
value  or  values  of  w,  if  any,  the  similar  formulse  which  we  shall  have,  do  not 
hold. 


200  CALCULUS  [Ch.  VI. 

whence,  by  integrating, 

(3)    tan  X  sec"~^  x  =  n  —  1  j  sec"  xdx  —  (n  —  2^  \  sec"~^  x  dx, 

/„      -,        tan  X  sec""^  x  ,  n  —  2  C      n-i     j 
sec"  xdx  = 1 I  sec"  ^  x  dx. 
n  —  \            n  —  1^ 

By  repeated  application  of  this  formula  of  reduction,  the 
determination  of  the  integral  will  be  reduced  either  to  that 
of  J  dx  or  of   I  secxdx,  both  of  which  are  known. 

There  are  many  varieties  of  formulae  of  reduction,  but 
the  scope  of  our  work  does  not  permit  us  to  take  up  even 
the  simpler  ones  of  them. 

Art.  7.      Integration  by  inspection.     I.    If  the  function 

J,  to  be  integrated  can  be  separated  by  inspection  into  two 

factors,  one  of  which  is  the  derivative  of  the  other,  then  the 

integral  is  equal  to  one-half  the  square  of  the  latter  factor. 

In  symbols, 

/du  J        u^ 
dx  2 

n+l 


and  more  generally,  \u^-—dx  =  — — -^ 

^       dx  n  +  1 


as  may  readily  be  proved  by  differentiating  the  right  mem- 
bers. « 

EXAMPLES 


.    ^{x^  +  2  x){Z  x^  +  2)  rfa;  =  (^^  +  ^  ^^ 

2.  f  sill  X  cos  xdx=  ^H^- 
J  2 

«      r  sin  a:    ,         f  sin  x         \       ■,        C ,  ^  o     j        tan^  x 

3.  I dx  =\ • ax  —\  tan  x  sec^  xdx  =  — - — • 

J  cos^  X  J  cos  X    cos^  X  J  2 


COS' 

1 


:.  j'.-(a^  +  x^)i</x=l2i±£!H. 


6-7.]  THE  SIMPLER  METHODS   OF  INTEGRATION  201 

II.  If  the  function  to  be  integrated  can  be  written  as 
a  fraction  whose  numerator  is  the  derivative  of  the  denomi- 
nator, then  the  integral  is  the  logarithm  of  the  denominator. 
In  symbols, 


/ 


du 

dx  ,        , 

—  dx=  iosf  u. 


EXAMPLES 

The  various  preceding  calculations  can  be  somewhat 
abbreviated  in  form  by  not  explicitly  introducing  the  new 
functional  symbol  u.  This  is  analogous  to  the  abbreviation 
spoken  of  in  the  Differential  Calculus  (p.  154),  and  the 
cautions  there  given  to  the  student  with  regard  to  using 
this  method  may  be  repeated  with  double  emphasis  here. 
In  addition  to  making   the  mental  transformation  of   the 

function  into  terms  of  u^  the  value  of  -—  must  be  mentally 

du 

computed  and  substituted,  making  a  process  of  several  steps 
in  which  mistakes  may  easily  occur.  The  student  should 
not  attempt  to  use  this  method,  even  in  simple  cases,  until 
he  is  quite  skilful  in  determining  the  integral  by  the  formal 
use  of  the  function  u. 

EXERCISES   XXII 


Show  that 


dx. 


Via: 


I.    {x^^a^  -x^dx  =  ^^  Va2  +  x^  +  -^  (*— ^ 
J  n  +  2  n-\-2J  y/a'2 

'ts,  then  use  the  formula 

\.    (x'^Va^  -h  x^dx  =  ^^  Va2  +  x^  +  -^—  f  -    

J  n  +  2  n  +  2J  Va^  +  x^ 


Hint  :    Integrate  by  parts,  then  use  the  formula  of  reduction  (1), 
p.  199. 

x^dx 


202  CALCULtrs  LUH.  VL 

Find  the  value  of : 


f       ^'       dx.     ■  18.    f  ^^^  , 


10 

11 

12 

13.    re^(e^  +  a)  (fa:. 


'^  \a  +  OX     a  —  oxj 
15.    r^^ccosrr^^ 


16 
17 


3x3 


5.  (—^—-dx,  19.  f— ^^^^^ dx. 

-^  Vl  -  a:2  ^  V9+a:2 

6.  (^ec'^xdx,  .            20.  jsecSxJx. 

7.  fa:3V4  -  a^^Jar.  21.  J  sec^  6  x  </a;. 

r      , o«  r  3  x'^dx 

a    JarV^^M:^^^.  22.  J  ^^j^jj^g 


^  3  -  5  a:  ^  vTT^^ 

J      5      ^^  24.    I  cos2a:sina:t?a:. 


loar  VI  +  x'K 


25.    \^m^xco?>xdx. 


*^Va:  +  2      x-2;  ,  ^ 

^  1  26.    \  cos**  a:  sin  a:  dx. 

*^       X  ram.  tn.n  -r   _ 


27.     t  — -"'"^^3,, 


Tare  tan  a 
*   J    1  +  a:2 

28.  p^-^-^J:r. 

29.  (1^:1:5255  rfx. 

•^  x  —  Sin  X 

30.  JO^g^)"rfa:. 
r  (2ax-a)dx  ^  ^ 

J{ax^-ax  +  l)  31.    J  (a:2  +  3  a;  -  l)3(2a:  +  3)(/a:. 

C    x^dx 
•   J^i— 73*     ,  32.   ^x-\x-^  +  5)-i^dx. 

Show  that : 

33.  fV^^3T2rfa:=^^^^iE«!_^'log(a:+V^?^3^). 
•^  ,  2  2 

34.  f  — -£! 6?a;  =  ^  Va:^  4-  r/2  -  £^  log  (x  +  Va:^  -<-  g^), 

35.  J  cos^ajrfa:  =  :|sin2a:  +  |a:. 

36.  j'sin2rfa:  =  -  1  sin  2  a:  +  1  a:. 


7-8]  THE  SIMPLER  METHODS   OF  INTEGRATION  203 

Art.  8.  Decomposition  into  partial  fractions.  The  inte- 
gration of  rational  fractions  is  usually  accomplished  by 
breaking  up  the  given  fraction  into  a  sum  of  simpler  frac- 
tions.    The  following  examples  will  make  the  method  clear: 


J  {a 


"^^  (h^a). 


x)(h  —  xy 


We  shall  show  first  that  numbers  p  and  q  can  be  found 
such  that 


{a  —  x)Q)  —  x)      a  —  x      h  —  x 
We  notice  that 

p  q      _bp  -^aq-(p-\-  q)x 


a  —  x      h  —  x  {a  —  x^{h  —  x^ 

If  p  and  q  be  chosen  so  that 

(2)  hp  +  aq=\,    p  +  q  =  0, 

the  numerator  of  the  fraction  last  written  has  the  value  1, 
and  the  equation  (1)  is  established. 

The  desired  values  of  p  and  q  can  always  be  found  by 
solving  the  system  (2)  for  the  two  unknown  quantities  p 
and  q^  with  the  results 

(3)  p  =  - ,     q  =  - 


h  —  a  h—  a 

We  find,  therefore,  that 

r        dx        ^  r  1        dx      r  i        dx 

^  (a  —  x)(h  —  x)     J  h  —  a      a  —  x     ^  b  —  a      h  —  x 

(4)  =_      l_log(cjj_a;)+--i-log(5-:r) 

0  —  a  0  —  a 

1  T      h  —  X  ,    ri 
log h  (7. 


h  —  a       a  —  x 


204  CALCULUS  ~  [Ch.  VI. 

II.    In  a  similar  manner,  the  integral 
A-^Bx 


f: 


dx^       (b  ^  a)^ 


(a  —  x}(^b  —  x) 

where  A  and  B  are  given  constants,  may  be  determined. 
Here,  also,  in  the  first  place  we  seek  to  determine  two  num-. 
bers  J)  and  q  such  that 

(5)  ^  +  -^^ =  _^  +  ^L_. 

(^a  —  x}(^b  —  x)      a  —  X      h  —  x 

Just  as  above,  we  find  that  if  we  can  determine  j9  and  g'  to 
satisfy  the  conditions 

(6)  jph  +  qa  =  A,    -(^2^  +  q^=B, 

the  relation  (5)  will  be  established.  By  solving  the  equa- 
tions (6),  regarding  p  and  q  as  unknown  quantities,  we  find 

.rrx  A^  Ba  A  +  Bh 

(7)  p  =  -T ->     q  = r- 

b  —  a  a—  b 

For  brevity,  we  shall  still  retain  the  symbols  p  and  q  in 
our  work,  they  having  now  the  values  just  found. 
.  We  have  then 

(8)  /'      -^  +  f     ^d.  ^fP^+f^ 
^  (a  —  x^{b  —  x)  J  a  —  X     J  b  —  x 

=  p  log +  q  log +  (7, 


where  p  and  g  have  the  values  found  at  (7). 

The  resolution  of  the  fraction  under  the  integral  sign  into 
the  sum  of  two  fractions,  whose  denominators  are  respec- 
tively the  two  factors  of  the  denominator  of  the  given  frac- 
tion is  known  as  Decomposition  into  Partial  Fractions.* 

*  This  problem  is  the  inverse  of  the  problem  to  reduce  given  fractions  to 
a  common  denominator  and  add. 


8.]  THE  SIMPLER  METHODS   OF  INTEGRATION  205 

The  following  are  special  cases  of  the  above  results : 

(10)'  =iogA^^+c.* 

III.  The  method  explained  above  can  be  extended  at 
once  to  the  case  that  the  denominator  of  the  fraction  under 
the  sign  of  integration  can  be  broken  up  into  more  than  two 
factors.  We  shall  treat  the  case  of  three  factors,  from  which 
the  method  to  be  followed  in  the  case  of  more  than  three 
factors  is  clear. 

We  consider  the  integral 


/; 


ax, 


(<2  —  a;)  (ft  —  x")(^c  —  x) 

where  A^  B,  (7,  a,  ft,  c,  are  given  numbers  (a,  ft,  c  unequal). 
We  seek  to  determine  three  numbers,  j9,  q^  r,  such  that 

(11)  A^-Bx+Cx'         ^     V      ^      q      ^      - 


(^a  —  x)(h  —  x^(c  —  x)      a  —  x      h  —  x      c  —  x 

shall  be  true  for  all  values  of  x.  This  could  be  done 
analogously  to  the  previous  examples  by  reducing  the  right 
member  to  a  common  denominator,  and  then  equating  the 
coefficient  of  x^  in  the  resulting  numerator  with  (7,  that  of  x 
with  B,  and  the  term  free  from  x  with  A.  Three  equations 
of  the  first  degree  would  result,  from  which  the  values  of 

*  Formulae  5  and  7,  Appendix. 


206  CALCULUS  [Ch.  VL 

the  three  unknown  quantities,  p^  q^  r,  could  be  determined. 
But  they  may  be  determined  also  by  the  following  method  : 
Multiplying  the  equations  (9)  by  a  —  a:,  we  obtain 

(12  )  —— ! ! =  p  -\-  q \-  r 

^     "^  (h-x)(^c-x)      ^      ^h-x        b-x 

Since  relation  (11)  is  to  hold  for  all  values  of  x^  the  rela- 
tion just  written  must  do  so  too,  and  we  have,  for  x  =  a  in 

particular 

A^-  Ba  +  CaP' 


{h  —  a){c  —  a) 
Quite  similarly  we  find 
^-^3^  ^^A±Bh±C^^      ^^A  +  Bc  +  Ce^ 


(a  -  ^)(c  ~b)  (a  -  c){b  -  c) 

For  brevity,  we  still  retain  the  symbols  jt?,  q^  r,  to  denote 
these  values,  and  have 

(14)      C-A±Bi±G2^_a, 

^  (^a  —  x}(b  —  x} (c  —  a;) 

/pdx        rqdx        rrdx 
a—x     Jh—x     Jc—x 

=  ;?  1% h  q  log h  r  log  — h  C. 

a  —  x  0  —  X  c  —  X 

Example :  To  determine 

(15)         r ^-^-+^-' dx. 

^     ^  ^  (1  -2:)(2-^)(3-a;) 

Here      A  =  1,  B  =  -2,   (7=3,  a  =  1,  5  =  2,  c  =  3. 
Accordingly,      jt?  =  l,    ^  =  —  9,   r  =  ll; 


8.]  THE  SIMPLER  METHODS   OF  INTEGRATION  207 

and,  consequently, 


(!*')  /; 


l-2a;  +  3:r2 


dx 


dx 


=  f  dx  _  r9dx     r lid 
J 1-x   J 2-x   J  ^- 


X 


logr^  +  9  log (2  -  a;)+  11  log-i-  +  C 
1  —  x  6  —  X 


=  W ^^  ~  ^^^ +  C'.* 

^(l-a:)(3-a;)ii 


IV.   We  take  up  next  the  case  in  which  two  of  the  factors 
of  the  denominator  are  equal. 

Let  there  be  given  for  integration 

(17)  (z ^T r,       (*^«)- 

*^  {a  —  xyi^b  —  X) 

We  put 

(18)  1  ^        P         I       ^      I       ''    , 
(a  —  x)\b  —  x}      (a  —  x^      a  —  x      h  —  x 

where  jt?,  ^,  r,  are  numbers  to  be  determined  ;   from  this  by 
clearing  of  fractions, 

(19)  1==  'p(h-x)+q(a-  x^ (h  -x)-{-r(a-  x^. 

This  equation  is  to  hold  for  all  values  of  x.     For  x  =  a, 
in  particular,  we  have 

l  =  p(h-a)    or  p  =  T , 

and  for  x  =  b,  1  =  r  (a  —  5)^    or    r  = —r, 

(a  -  6)2 

and  for  a;  =  0,  1  =  pb  +  q  ab  -{■  ra\ 

*  Forniulse  5  and  7,  Appendix. 
15 


208  CALCULUS  [Ch.  vl 


whence,  by  substituting  1 
find  that 

]he  values  found  for 
1 

P 

and  r, 

1 

we 

9  — 

Therefore 

1 

(^a-by 

1       1,1 

{a  -  x)\b  -  x) 
1                1 

b-a      (a-xy      ib 
and 

-ay(a-x^      {b- 

ay 

=      (6- 

X) 

xy(b  -  x) 

dx 


^     1      r    dx i__  r  dx  1      r 

(b-a)  J  {a-  xy      (b  -  ayJ  a  -  x      {b  -  ay  J  b 
1  1  1        1  1       ,         1        ,  1 

log +  71 r2  log 


X 


b  —  a      a  —  X      (^b  —  ay        a  —  x      (b  —  ^y        b  —  x 

1  1  1        ^      b  —  X  ,   ^ 

log +  (7. 


b  —  a      a  —  X      (b  —  o,y        a  —  x 
V.    We  treat  next  the  integral 

(21)  /r-^M#-^'^^'   (**«)• 

•^  (a  —  xy^b  —  X) 
Here,  likewise,  we  put 

(22)  -^  +  f    ,-^^2  +  ^  +  r^ 

{a  —  xy{b  —  x)      {ct  —  xy      a  —  X      b  —x 
and  have,  by  clearing  of  fractions, 

(23)  A  +  Bx  ^  p(b  —  x)+  q{a  -  x)(b  -  x)-^  r{a  -  xy  \ 

whence,  putting  x  equal  to  a,  6,  and  zero  in  turn,  we  find  that 

^A  +  Ba         ^  A-{-Bb  ^      A  4- Bb 

^        b-a'      ^      (a  -  by'     ^  (a-  by 


8.]  THE  SIMPLER  METHODS   OF  INTEGRATION  209 

Accordingly, 
(24)  r A  +  ^?^ dx- 


rA±Ba  ^       dx  CA  +  Bb  ^  _d^,    CA  + 

J    h  —  a       (a  —  xf'    ^  {a  —  hy'     a  —  x     ^  («  — 


Bb       dx 


by   b-x 


^+Ba,J, A  +  Bb        _J_^A  +  Bb        J^ 

b  —  a       a  —  x      (a  —  by        a  —  x      {a  —  by        b  — 

A-{-Ba     1      ^   A-^Bb.a-x  ^  ^^ 
b  —  a    a  —  x      i^a  —  by        b  —  x 

Bx-\-i 

+  bxy 


Put 

A  +  Bx±C^  _        p 


^     ^         (a  +  bxy  (a  +  bxy  '   (a  +  bxy  '   (a  +  bx} 

Clearing  of  fractions, 
(26)  A  +  Bx-^Cx^=  p  +  q(a  +  bx}  +  r{a  +  bxy. 


Put  X  =  —  -'> 
0 

Put  x=0, 


.      Ba  ,  Ca^ 


A=p-^qa-\-  ra\ 
Put  x=l, 

A-\-B-\-0=p-^-q(a-}-b:i-h  r(a  +  5)2. 

Substituting  the  value  of  p  found  above  in  the  last  two 
equations  and  solving,  we  find  the  values  of  q  and  r,  viz., 

B     2aC 


<1  = 


r  = 


b  62 

(72 


210                                              CALCULUS  [Ch.  VL 

Accordingly, 

.^rjr.       CA  +  Bx  +  6V  ^  r      y?6?a^           r      q  dx  r_r 

^"  ^     J      (a  +  6:r)3         -^  (a  +  62:)3  "^  J  (a  -|  hx^  -^  a  - 


-\-bx 


=  26(^^  +  60^  +  ^'°^^"  +  *^^  +  '''' 

where  jt>,  5',  r  have  the  values  found  above,  and  C  is  added 
as  the  arbitrary  constant  of  integration,  to  avoid  confusion 
with  the  (7  given  in  the  integral. 

VII.  We  have  found  the  unknown  quantities  by  substi- 
tuting special  values  of  x.  We  could  also  .  find  them  by 
making  use  of  the  principle  of  algebra,  that  if  two  expres- 
sions are  equal  for  all  values  of  x^  the  coefficients  of  the 
different  powers  of  x  in  one  expression  are  respectively 
equal  to  the  coefficients  of  the  same  powers  of  x  in  the  other 
expression. 

We  determine  by  this  method  the  integral 


Pu 

(29) 


(a: -1)3(2  a; -j- 7) 
Put 


(x-iyQlx  +  1) 


•      ix-iy      (x  -ly      {x-1)      (2x  +  7) 

Clearing  of  fractions,  multiplying  out  the  right  member, 
and  collecting, 

(30)    10a^-nx^-4.bx  +  m 

=  (2  r  +  s)^  +  (2  ^  +  3  r  -  3  s)a:2 

+  (2  ^  -h  5  ^  -  12  r  +  3  s)a:  +  (7  J9  -  7  g  -h  7  r  -  s). 


8.]  TH^  SIMPLER  METHODS   OF  tNTEGttATlON  211 

Equating  coefficients  of  like  powers  of  a;, 
10  =  2  r  +  s, 
-13  =  2g  +  3r-3s, 

(31)  -45  =  2jt?  +  5^-12r  +  3s, 

m  =  l  p-1  q  +  1  r-s. 

Solving  by  the  methods  of  elementary  algebra,  we  find 
jo  =  2,  ^  =  —  5,  r  =  3,  s  =  4. 

These  values  could  have  been  found  more  readily  by  com- 
bining the  two  methods  as  follows  : 

Clearing  the  equation  (29)  of  fractions,  we  have 

(32)  10  a;3_i3^2_45^  + 66=^(2  a;4-T)+^(2:r  +  7)(a^-l) 

+  r(2^  +  7)(x-l)2  +  s(a:-l)3. 

In  this  put  a;  =  1,  and  thus 

18  =  9jt?,  or  j9  =  2. 
Similarly,  putting     x—  —'^^ 

-  2.^±^=  - 1^  s,  or  s  =  4. 
Equating  coefficients  of  o(^  and  x^  in  (30),  we  have 
10  =  2  r  +  s, 
-13  =  2^  +  3r-3s, 
whence  r  =  3,  ^  =  —  5. 

We  have  not  made  use  of  the  coefficients  of  the  first  power 
of  x^  and  of  the  term  free  from  x.  They  give  rise  to  the 
last  two  of  equations  (31),  and  must  be  satisfied  by  the 
values  of  p^  q^  r,  «,  which  we  have  found.  This  affords  a 
check  against  numerical  mistakes  in  the  calculation. 


212  CALCULUS  [Ch.  VI. 

Returning  now  to  our  integral,  we  find 

=  -(2;-l)-2  +  5(2^-l)-i  +  31og(a:-l)  +  21og(2a;  +  7)  +  a 

VIII.  In  all  the  examples  which  we  have  treated,  the 
degree  of  the  numerator  of  the  fraction  given  for  integra- 
tion has  been  lower  than  that  of  the  denominator.  If  this 
should  not  be  the  case,  the  fraction  given  can,  by  ordinary 
division,  be  expressed  as  the  sum  of  a  polynomial  and  a 
fraction  whose  numerator  is  of  lower  degree  than  its 
denominator,  and  the  methods  of  this  paragraph  can  be 
applied  to  the  latter  fraction.  A  single  example  will  illus- 
trate the  process  sufficiently, 


(34)  / 


x^-'6x-\-'2 


dx. 


2a;4_62:3_^2_pl8^_3     ^^2_5+        ^X+1 


x^-Sx-^2  x^-Sx-{-2 

By  the  method  of  partial  fractions,  we  find 

r35^  3a^  +  7      ^    13  10 

^     ^  x^-Sx-{-2     x-2     x-i 

Accordingly, 

^     ^    J  x-^-'6x  +  2 

=  ^a^-3x-\-mogCx-2}-10logCx-l)  +  C. 

The  examples  which  we  have  solved  will  suffice  to  show 
the  student  how  to  apply  this  method  in  all  the  cases  that 


8-9.]  THE  SIMPLER   METHODS   OF  INTEGRATION  213 

will  arise  in  the  course  of  our  work.  For  an  a  priori 
demonstration  that  it  is  always  possible  to  determine  our 
numbers  jt?,  q,  r,  •••,  in  one  and  only  one  way,  and  for  the 
treatment  of  more  complicated  cases  we  refer  to  standard 
works  on  algebra  and  to  more  extended  treatises  on  the 
Integral  Calculus.  These  works  will  also  give  many  other 
methods  and  artifices  for  integration,  and  an  abundance  of 
exercises  for  practice,  more  difficult  and  complicated  than 
those  we  have  undertaken. 

EXERCISES    XXIII 

Integrate  by  decomposition  into  partial  fractions : 

^^    r    (x-l)dx    ,       5    r  7-2x  ^^^  ^     r_dx — ^ 

J  (x-'d)(x  +  2)  Jx^(7-x)  Jx^-Qx  +  5 

2.  f "^ 6.    r3x2-5a:  +  ll^^^        10     C dx 

J  (x  -\-  d){x  -  4:)  J      (1  +  x)3  J2x-3  x^ 

3.  f-i^.  7.    (-l_dx.(a>x).      11.     f  ;^^  +  -     dx. 
^  1  —  X-'           ^  J  d^  —  x^  J  x^  —  X  —  2 

.    f-^-  8.    C^—dx.(a<x).      12.     (^±ldx. 

•^  1  —  a:*  -'  x'^  —  d^  *^  x^  —  X 


4 


13.  r  ^x^-^^x+^^u   ^^^ 
^0   ~ 


[x-3Xx  +  S)(x-Q) 


Art.  9.  Summary  of  results.  We  collect  into  a  table 
the  most  important  integrals  which  we  have  determined, 
including  some  from  the  lists  of  exercises.  For  ease  of 
reference,  the  integrals  are  grouped  according  to  the  func- 
tion to  be  integrated.  The  constant  of  integration  is 
omitted  throughout. 

Exercise.  As  the  integrals  are  arranged  neither  in  order  of  difficulty 
nor  according  to  methods  of  proof,  it  would  be  an  excellent  review  to 
regard  the  table  as  a  set  of  miscellaneous  exercises  for  proof. 


214  CALCULUS  [Ch.  VI. 

TABLE  OF  INTEGRALS 
I.  General  formulae. 

^1.    \{u  +  v  +  w^  ...)  dx  =  \udoc-\-  \vdx  +  \wdx  +  .... 

^2.    \audx  =  aiu  dx. 

a.    (fix)  dx  =  (fix)  —  du. 
J  J  du 


du 


t"   J  -^  dx  —  log  u 


u 


6.    fi*-^f«^=^ 
J       dx  n-\-\ 

II.  Rational  algebraic  functions. 

6.    (x''dx=^^~—* 
J  n4-  1 

8.  r^= logic. 

»i/ T ^^  =  i-  arc  tan  ^  =  -  i  arc  cot  5.      , 

10.    f_^^  =  J-log^^. 

^  x^  -  d^     2  a         ic  +  a 

III.  Irrational  algebraic  functions. 


\^(ia  ±  x)""  dx  =  ±  ^^  ^  ^^"^ 

14.  (*  Va^  -  x^  <Zx=  ^  Va2  _  ^2  +  ^  arc  sin  ^. 
»/  2  2  a 

15.  J  V^^  -a^dx=^  Vx^  _  a2  _  f^  j^^  (^^  ^ y^2  _  ^2-)^ 

16.  r__-^_-z=iog(i;c+V^2^^). 
*^  Vx'^  ±  a2 


0.]  THE  SIMPLER  METHODS  OF  INTEGRATION  216 

17.    f       <i^       =  arc  sin '^.        '-"' 


18.    C_jJLjto_^  +  v„a^ 


Va"  ±  a? 


19      , 


20.    C       ^^        dx  =  -f  Va^  -  sc'^  +  ^  arc  sin  -• 


^      J^ -ijc'^  +  ^  arc  sin^' 

Va'-a^^  2  2  a 


IV.  Trigonometric  functions. 

21.  (  sin  oc  doc  =  —  cos  oc, 

22.  r cos  acdQc  =  sin  a?. 

23.  Itan  a?  ciac  =  -  log  cos  x, 

24.  ( cot  X  dx  =  log  sin  x. 

25.  (4^  =  log  tan  f. 
»/  sm  a?  2 

,  26.    f  ^^^  =  log  tan  (f  +  '^X 
P^^  J  cos  a?  V2      4/ 

27.  f-^  =  -cotic. 
*^  sin"  a? 

28.  f-^=tanic. 

'^  cos'"  a? 

29.  Jsin  i^c  cos  ic  f/x  =  ^^?^. 

30.  f  .     ^^^ ^logtanic. 

-'  sin  a?  cos  x 

31.  I  ar;  sin  a?  fiat;  =  -  a?  cos  x  +  sin  a?. 

32.  (  x^  sin  a?  dx  =  -  a?^  cos  a^  +  2  a?  sin  a?  +  2  cos  x, 

33.  ( cos^  a?  fZa?  =  ^  sin  2  a?  + 1  a?. 

34.  J  sin^  X  dx  —  -^^m  ^  X  ^\  X. 


216  CALCULUS  [Ch.  VI.    ** 

V.  Logarithmic  functions. 

35.  Jlog  X  dx  =  30  \o^  oc  -  nc, 

36.  r X-  log  xdx=  ^"^'^(^  +  ^)  ^<>g  ^  -  11. 

yi.  Exponential  functions. 

37.  (e^dx^e"". 

38.  ra*f?ic=-^^. 
J  log  a 


39.    Cire^^rfic 

J  ^2 


e"^(^ta;-l)^ 


VII.  Formulae  of  reduction. 

40.    (ii  —  dx  =  tiv  -  iv  ^-  dx. 
J      dx  J     dx 


J  » +  2  n  +  2J  ^^i  _  ^2 

44.  fx»V^^T^^dx  =  ^V^^Tl?  +  ^f^^^. 

45.  fsec''x«?x  =  <^°"'^''''"''^  +  g^fsec"-^a;rfa;. 

J  W.—  1  M-l»/ 

EXERCISES  XXIV  (MISCELLANEOUS) 
Show  that : 

1.  f  (2  +  7  xydx  =  J  (2  +  7xy. 

2.  f ^ dx  = L_^,  (put  2  + 3^:2  =  m). 

J('2  +  3x^y  4(2  +  3  a;^)-^ 

3.  (   'I^ =f '^ -arctan(x-4). 


90  THE  SIMPLER  METHODS  OF  INTEGRATION  217 


J  _  15  4.8a;- x^      Jl_(a:_4)2  ^^5-a: 

6.  'f ^^^ ^-l-log^  +  2-V2. 

^a:2  +  4a:  +  2      2V2     %  +  2  +  V2 

7.  f ^^?:==(l+a;)[|VrT^-l]. 

•^  !•+ vl  +a; 

„     C  sin  a:    , 

8.  \ ax  =  sec  x. 

•^  cos^  a; 

^-  i'(x-lK^H4) "^ '"^ ^^"^^ ~"^^°^ (a^H4)  + j arctan |. 

10.  i  COS  (jjx  +  q)dx  =  -  sin  (j9x  +  q). 
J  p 

11.  I  sec^  xdx  —  tan  a:  +  |  tan^  ^  +  i  tan^  a:.  , 

12.  f(l-cosa:)2rf:r=^ilL?^_2sina:  +  — . 
J  4  2 

13.  i  ^_^  =  —  a:  cot  .r  +  log  sin  x. 

J  sin^a: 


14.     \  ~ =  x  tan  X  +  W  cos  a:. 


r  a:r/a 
'J  cos^ 


cos^  x 

15.  r^^^:=2vS^. 

Vsinx 

/*  •  SI  n   IT 

16.  I  cos^x  Ja:  =  sin  a: . 

J  8 


18.  J  (loga:)Va;  =  a:(loga:)2- 2  a:  loga:  +  2  a:. 

19.  rx2V^+2^7a:=?i:^±fI'(15a:2-24a:  +  32).     Put  V^rr2  =  M. 
*^  105 

on      fl-^^  /         aF^^  .  ^'^      a;3  ,  a:i      a:^   ,      l      .       /  i   ,  i\1 

Put  a:^  =  M. 


CHAPTER   VII 

SOME   APPLICATIONS   OF   THE   INTEGRAL   CALCULUS 

Art.  1.  The  attraction  of  a  rod.  If  two  material  points 
of  mass  m  and  m',  respectively,  are  at  a  distance  r  from  each 
other,  Newton's  Law  of  Gravitation  tells  us  that  an  attrac- 
tion A^  whose  amount  is 

(1)  A=-f, 

exists  between  them.  With  this  fact  given,  we  take  up  the 
problem : 

To  determine  the  amount  of  attraction  exerted  hy  a  HtraighU 
homogeneous  rod  of  uniform  thickness  and  of  length  I  upon 
a  material  point  P  of  mass  m,  which  is  situated  in  the  liiie 
of  direction  of  the  rod  and  at  the  distance  a  from  its  nearer 
end. 

We  take  the  length  of  the  rod  as  the  variable  and  denote  it 
by  X.  Since  the  attraction  depends  upon  the  value  of  x,  we 
let  F(x^  denote  the  function  that  we  wish  to  determine. 
We  now  ask  ourselves  by  how  much  the  attraction  is  in- 
creased when  the  length  of  the  rod  is  augmented  by  h  at 
its  further  end,  the  distance  between  the  nearer  end  of  the 
rod  and  the  point  remaining' constant.  When  the  length 
of  the  rod  is  increased  by  h  the  total  attraction  becomes 
F(^x  +  A),  and  the  attraction  of  the  added  piece  of  length 
h  is  therefore  F(x  -\- h)  —  F(x),  Let  us  suppose  now  this 
added  piece  to  be  replaced  by  a  material  point  of  equal  mass 

218 


1.]  APPLICATIONS   OF  INTEGRATION  219 

situated  where  it  will  exert  the  same  attraction ;  this  point 
will  be  somewhere  between  the  two  ends  of  the  added 
piece.  If  M  denote  the  mass  of  the  unit-length  of  the  rod, 
and  if  e  denote  the  distance  of  the  material  point  from  that 
end  of  the  rod  which  is  further  from  the  point  P,  we  have, 
by  Newton's  Law,  that  the  added  piece  exerts  on  P  the 
attraction 
^ON  mMh 

We  have  thus  found  two  different  expressions  for  the 
attraction  of  the  added  piece.     Equating  these,  we  have 

^  ^         ^  ^      {a  +  x  +  ey 

Fix  +  h) -  F(x)  ^         mM 

h  (a  +  a;  +  e)2 

This  holds  true  for  every  length  h.  When  h  approaches 
the  limit  zero,  the  limit  of  the  left  member  is  the  derivative 
of  F(x).  In  the  right  member  e  also  approaches  zero  since, 
by  definition,  e  is  positive  and  less  than  Ji.     Hence, 

.^^  dF(x)  ^      mM 

dx  (a  +  xy' 


Integrating,  F{x)  =1  — ''- 


mM      n 
dx. 


(a  +  xf 
This  integral  is  very  easily  found,  and  we  have,  finally, 

(4)  A  =  F(x)  =  -^^^^-G. 

a  -\-  X 

It  is  apparent  that  a  rod  of  length  x  =  0  exerts  the  attrac- 
tion, ^  =  0.     If,  in  equation  (4),  x  be  put  equal  to  zero, 

(5)  ■  0  =  -!^+(7, 


220  CALCULUS  [Ch.  vil 

and  by  subtraction  of  (5)  from  (4), 


(6)  •  A  =  mM(-  - 


1 


\a      a  -\-  X. 

The  attraction  of  the  rod  of  length  I  is  found   by  the 
substitution  of  I  for  x  in  the  last  equation;  then 

(7)  A  =  mM(^ 


a  +  l 


If  the  rod  be  very  long,  the  fraction  — —  becomes  very 

small,  so  that  it  may  be  neglected  *  in  comparison  with  — 
The  equation  then  assumes  the  simple  form 

(8)  4  =  !^. 

a 

A  rod,  then,  whose  length  is  very  great  in  comparison  with 
the  distance  of  one  of  its  ends  from  a  material  point  l,ying 
in  the  prolongation  of  its  axis,  exerts  an  attraction  wJiich  is 
practically  independent  of  its  length  and  iiiversely  propor- 
tional to  the  distance  of  the  point  from  its  nearer  end. 

Art.  2.  The  hypsometric  formula.  What  is  the  atmos- 
pheric pressure  at  the  height  H  above  the  earths  surface  ? 

Let  the  pressure  (in  centimeters  of  mercury)  and  the 
density  (referred  to  mercury)  of  the  air  at  the  earth's  sur- 
face be  denoted  by  B  and  S  respectively.  Atmospheric 
pressure  being  due  to  the  weight  of  the  air,  a  column  of  air 
H  cms.  high  and  of  the  uniform  density  aS',  exerts  a  pressure 

*  In  mathematical  considerations  we  seek  exact  results,  and  never  neglect 
anything  ;  in  physical  considerations,  where  the  results  are  at  best  only 
approximately  accurate,  it  is  often  permissible  to  neglect  quantities  small 
enough  not  to  affect  the  result  appreciably. 


1-2.]  APPLICATIONS   OF  INTEGRATION    .  221 

equal  to  ^aS'  cms.  of  mercury  ;  and  if  the  air's  density  were 
the  same  at  all  elevations,  its  pressure  would  diminish  aS'  cms. 
of  mercury  for  every  centimeter  of  elevation.  But,  in 
reality,  the  density  of  the  air  also  diminishes  with  a  decrease 
of  pressure ;  and  in  accordance  with  Boyle's  Law  (p.  3) 
the  density  and  the  pressure  of  the  air  are  directly  propor- 
tional to  each  other. 

Let  the  pressure  at  the  height  x  above  the  surface  of  the 
earth  be  denoted  by  F(x)^  and  the  density  by  s.  Then  at 
a  higher  elevation  a: +  ^,  (the  density  at  which  maybe  denoted 
by  s',)  the  pressure  would  be  diminished  by  sA,  if  throughout 
the  additional  elevation,  the  density  were  constantly  s.  But 
this  is  too  great  as  the  density  decreases  from  s  as  its  largest 
value.  On  the  other  hand,  it  would  be  diminished  by  s'h 
if  the  density  were  constantly  s';  but  this  is  too  little, 
since  s'  is  the  least  density  in  the  additional  elevation. 
The  true  value  must  therefore  be  intermediate,  as  (s  —  e)A, 
where  0  <  €<  s  —  s\     We  accordingly  have 

(1)  F(x  +  A)  -  F(x^  =  -  («  -  6)^, 

the  right  member  being  affected  with  the  minus  sign,  inas- 
much as  the  pressure  decreases  with  increased  height. 
According  to  Boyle's  Law, 

(2)  s:S=  F(x^  :  B, 
whence  s  =  jS  —^^' 

Combining  this  equation  with  (1),  we  have 

F(x  +  A)  -  Fix-)  =  -  (^I^  -  e\h, 

„.      F(x-\-K)-F(x)   _     fSF(x)       \           SF(x-)^ 
or  (3) --[—^ '-' -Y~ 


222  .  CALCULUS  [Ch.  VII. 

Allowing  h  to  approach  the  limit  zero,  and  bearing  in 
mind  that  when  h  approaches  zero,  s'  approaches  s,  and 
hence  e  approaches  zero,  we  have 

dFijx^  ^  _  SF(x)  . 
dx  B      ' 

and  putting,  for  brevity, 

Fix)=y, 

dy^_§y, 

dx  B 

Therefore  (p.  142), 
fA.\  dx  _  _  B 

^^  d^~     ¥ 

and  integrating  with  respect  to  y, 

(5)  ^=J-§<^' 

Now  when  a?  =  0,  that  is,  at  the  earth's  surface,  y  =  B; 
substituting  in  (5), 

(6)  0=-|log£  +  C 

and  subtracting  (6)  from  (^^,  we  have 


^ 


x  = 

For  the 
x  =  H^  we 

(7) 

required 
have 

atmospheric  pressure  y 

at  the  elevation 

-U 

or  y  =  Be 


2.]  APPLICATIONS   OF  INTEGRATION  223 

Further,  by  transforming  equation  (7)  we  may  calculate 
the  height  H  above  the  earth's  surface  from  the  observed 
atmospheric  pressure ;  thus, 

(8)  -^=f>«^f 

this  is  the  so-called  hypsometric  formula. 

In  the  above  discussion  we  have  neglected  the  influence 
of  temperature,  moisture,  and  the  latitude  of  the  station; 
these  factors  naturally  complicate  the  solution  greatly.  To 
take  them  into  account  for  average   conditions,  the  factor 

—  may  be    replaced   by  1,820,000 ;    this   includes   also   the 

multiplier'  necessary  to  pass  to  common  logarithms ;  the 
formula  then  becomes 

(9)  2r=  1,820,000  logi(,|. 

This  formula  gives  the  height  in  cms.  when  the  barometer 
is  read  in  cms.  To  exemplify  its  use,  let  it  be  required  to 
ascertain  the  elevation  above  sea  level  at  which  the  barom- 
eter would  stand  at  1  cm.  In  this  case,  B  =16  and  y  =  1 ; 
then 

ir=  182  .  10* .  log  76  =  182  •  10* .  1.88  =  3,421,600  cms. 

EXERCISES   XXV 

1.  A  ball  whose  mass  is  1  gram  is  placed  at  a  distance  of  10  cms. 
from  a  homogeneous  rod  of  mass  1000  grams  and  in  direct  line  with  it. 
Wliat  is  the  attraction  between  the  ball  and  the  rod  if  the  length  of 
the  latter  is  (i.)  10,  (ii.)  1000,  (iii.)  lO^o  cms.,  respectively? 

2.  At  what  height  in  kilometers  (1  kilometer  =  100,000  cms.)  is  the 
pressure  of  the  atmosphere  equivalent  to  that  of  a  column  of  mercury 
1  micron  (1  micron  =  0.0001  cm.)  high?  (Such  a  pressure  of  mercury  is 
about  that  obtainable  in  a  very  good  vacuum.) 

3.  What  is  the  average  reading  of  the  barometer  at  a  station  5  kilo- 
meters above  sea  level  ? 

10 


224  CALCULUS  [Ch.  VII. 

Art.  3.  Newton's  law  of  cooling.  Giveri  a  body  at  the 
temperature  6^;  the  temperature  of  its  surroundings  being 
lower  and  constantly  equal  to  0^^  it  is  required  to  find  the  law 
according  to  which  the  temperature  of  the  body  falls. 

We  assume,  with  Newton,  that  the  rapidity  with  which 
the  body  loses  heat  depends  upon  the  nature  of  the  body, 
and  is  proportional  to  the  excess  of  its  temperature  over 
that  of  its  environment. 

Let  W(f)  denote  the  total  amount  of  heat  given  out  by 
the  body  from  the  beginning  of  cooling  up  to  the  time  ^, 
and  W(t  -\-  A)  the  amount  of  heat  given  out  up  to  the  time 
(^  4-  A) ;  tlien  W(t  +  A)  —  W(f)  will  represent  the  heat  given 
out  during  the  interval  of  time  h. 

Let  the  temperatures  at  the  times  t  and  t  +  hhQ  6  and  0'^ 
respectively.  Then  if  the  temperature  remained  the  same  as 
at  the  beginning  of  the  interval  A,  during  that  interval  an 
amount  of  heat  equal  to 

(1)  TFi(0  =  K^-^o)^ 

would  be  given  out,  where  A;  is  a  constant  multiplier,  (factor 
of  proportionality,)  dependent  for  its  value  upon  the  nature 
of  the  body.  W^  is,  however,  too  large,  since  the  tempera- 
ture falls.  Likewise,  if  the  temperature  remained  constant 
as  at  the  end  of  the  interval,  the  heat  given  off  would  be 

(2)  W^(t)  =  kiff-e^)h; 

but  this  is  too  small  since  6'  is  the  lowest  temperature  in  the 
interval.  The  true  amount  of  heat  being  thus  greater  than 
the  latter  and  smaller  than  the  former,  is  given  by  the 
expression 

(3)  W(f)  =  k(e  -0^-  €)A, 
where  0<e<e  -6', 

We  saw  also  that  the  heat  given  out  is 


I 


3.]  APPLICATIONS   OF  INTEGRATION  225 

Equating  these  two  expressions,  we  have 

Wit  +  h)  -  WQ)  =  K^  -  ^0  -  0^' 
or    "         m±J}l^m^=kie-e,-ey 

III 

The  limit  of  this  as  li  approaches  zero  is  (since  at  the  same 
time  Q^  approaches  Q  so  that  e  approaches  zero) 

(4)  ^=*(^_^^). 

This  equation  gives  us  the  derivative,  with  respect  to  ^,  of 
the  function  which  expresses  the  heat  given  out  in  terms  of 
the  time ;  but;  as  the  right  member  is  not  expressed  in  terms 
of  ^,  we  cannot  find  the  function  itself  {i.e.  integrate) 
directly.  The  following  considerations,  however,  enable  us 
to  utilize  this  equation  to  attain  the  desired  result;  viz,  to 
find  the  relation  between  the  time  which  has  elapsed  and  the 
temperature  at  that  time. 

The  amount  of  heat  W(€)  given  out  up  to  a  certain  time  t 
may  be  regarded  as  a  function  of  the  temperature  Q  at  that 
time  as  well  as  of  the  time  t  itself.  In  particular,  the  total 
amount  of  heat  given  off  by  the  body  in  cooling  down  from 
the  temperature  0  (which  may  be  taken  as  a  variable)  to  the 
temperature  of  the  environment  ^q  is  * 

(5)  TT-  mc{e^  -  (9), 

*  When  it  is  more  convenient  (and  can  be  done  without  confusion),  a 
gain  in  simplicity  of  notation  is  often  made  by  omitting  to  indicate  the  in- 
dependent variable  in  the  functional  symbol.  Thus,  instead  of  writing  W(t) , 
we  may  write  simply  W.  If  t  =  <p{d),  we  have  W(t)  =W{(f>{d)],  and  for 
this  also  we  may  write  simply  W^  provided  no  confusion  arises  through 
doing  so ;  the  context  indicating  whether  W  is  to  be  regarded  as  a  function 
of  t  or  of  d. 


226  CALCULUS  [Ch.  vii. 

where  m  denotes  the  mass  of  the  body  and  c  its  specific 
heat.* 

Differentiating  this  equation  with  respect  to  ^,  we  have 

(b)  ^=-^^'- 

But  6  itself,  i.e.  the  temperature  of  the  body  at  any  instant, 
is  a  function  of  the  time  t  which  has  elapsed  since  the  cool- 
ing began.     W  is  therefore  a  function  of  a  function,  and 

dW  dt  ^dW^ 
dt    de       dO' 

dW  dW 


or 


dt       de       dO       dt 


dd      dW_     dt      dW 
dt  dd 


Substituting  from  equations  qp  and  (6),  we  have 

^  ^  de~      kCd-do)'    dt  mc^  ^^' 

Before  integrating,  we  remark  that  the  mass  m  is  entirely 
independent  of  the  temperature  while  the  specific  heat  c  is 
very  nearly  so ;  regarding  them  both  therefore  as  constants 
and  integrating  equation  (7),  we  obtain 

,Q.  mc  r    dO 

or      (9)  t^~f\og{e-e,~)  +  0. 


*  The  specific  heat  of  any  substance  is  defined  to  be  the  ratio  between 
the  amount  of  heat  given  out  by  a  certain  mass  of  it  in  cooling  off  through 
one  degree  of  temperature,  to  the  amount  of  heat  given  off  by  an  equal  mass 
of  vi^ater  cooling  through  the  same  temperature  interval.  The  variations  of 
specific  heat  with  the  temperature  may  be  neglected  in  this  problem. 


3.] 


APPLICATIONS   OF  INTEGRATION 


227 


Now  when    ^  =  0,    then    6  =  0^;    on    substituting    these 
particular  values  in  equation  (9),  we  have 


(10) 


me 


\ogid,-e,-)=o. 


and  after  subtracting  (10)  from  (9),  we  get 


(11) 


^log^ 


Or 


d. 


We  have  derived  equation  (11)  by  the  aid  of  a  hypothesis 
which,  although  it  seems  in  itself  highly  probable,  may  yet 
be  subject  to  question.  We  cannot,  however,  test  it  directly, 
since  it  is  impossible  to  determine  experimentally  the  varia- 
tions of  temperature  in  very  short  intervals  of  time.  Yet 
we  may  ascertairj  ^with  considerable  accuracy  whether  or 
not  the  equation  is  in  correspondence  with  experimental 
facts.  Thus,  Winkelmann  *  made  a  series  of  observations, 
tabulated  below,  on  the  cooling  of  a  body  when  the  tempera- 
ture of  the  environment  was  constantly  kept  at  0°  C.  and 
the  initial  temperature  of  the  body  was  19°. 90  C.  ;  i.e. 
6>o=0°.00;    (9i  =  19°.90. 


e 

t 

Xzi: 

18°.9 

3.45 

0.006490 

16^9 

10.85 

'     0.006540 

14°.9 

19.30 

0.006509 

12°.9 

28.80 

0.006537 

10°.9 

40.10 

0.006519 

8°.9 

53.75 

0.006502 

6°.9 

70.95 

0.006483 

*  Wiedemann's  Annale'n  der  Physik,  Vol.  44,  p.  195.     1891. 


228  CALCULUS  [Ch.  VII. 

The  data  given  in  the  third  column  are  computed  from 
the  observed  values  of  t  and  6.  From  (11),  by  a  little 
transformation,  we  have 

(12)  0.4343  A  =  1  w.  ^i~^o 

(0.4343  is,  approximately,  the  modulus  of  Briggean  loga- 
rithms). 

As  the  left  member  consists  of  constant  quantities,  the 
right  member  must  likewise  be  constant,  which  is  seen  above 
to  be  the  case  ;  the  slight  and  irregular  variations  which 
occur  are  to  be  attributed  to  errors  in  the  observation  of  the 
various  values  of  the  time  and  temperature. 

The  foregoing  discussion  may  be  regarded  as  a  typical  ex- 
ample of  the  introductory  considerations  of  pp.  166  and  167, 
a  hypothesis  concerning  a  natural  phenomenon  receives  a 
mathematical  formulation,  and  thus  leads  to  the  establish- 
ment of  an  expression  containing  derivatives  (a  "  differential 
equation").  But  in  order  to  compare  the  requirements  of 
the  hypothesis  thus  formulated  with  actual  facts,  in  other 
words,  to  test  our  hypothesis  by  observation  and  experiment, 
we  have  to  deduce  by  integration  an  equation  that  is  freed 
from  derivatives  and  contains  only  finite  quantities  which 
are  directly  accessible  to  experiment  and  observation. 

Successful  setting  up  of  the  differential  equation  depends 
upon  the  acumen  of  the  investigator ;  thereafter,  its  integra- 
tion is  entirely  a  matter  of  mathematical  calculation. 

Exercise.  In  another  series  of  observations  with  the  temperature  of 
the  environment  at  0°  C,  Winkehnann  obtained  the  following  results, 
the  initial  temperature  being  14°.86  C. : 

6  =  14°.38        13°.42        12°.44        11'^.45        10°.26        9^97 
t  =  130  405  703  1026  1197  1570 

1  B  —  B 

Calculate  the  values  of  -  log,Q  ^ — ^'     ' 


3-4.]  APPLICATIONS  OP  iNTEGttATlON  ^29 

Art.  4.  Concerning  the  general  method  of  all  these  applications. 
The  quantity  c  which  has  been  used  in  each  of  the  preceding  problems 
is  quite  an  important  auxiliary  in  making  the  mathematical  formu- 
lation of  the  physical  facts.  We  know  two  values  giving  the  changes 
which  would  occur  if  the  change  proceeded  uniformly  (on  two  different 
hypotheses)  throughout  the  interval  h,  which  is  without  restrictions  as 
to  size  (in  particular,  the  interval  h  is  by  no  means  supposed  to  be 
small).  We  know  that  one  of  these  values  is  too  large  and  the  other 
too  small,  and  that  the  true  value  is  obtained  by  increasing  or  dimin- 
ishing one  of  the  factors  by  a  quantity  which  we  call  €.  We  know  that 
a  quantity  c  exists  which,  when  introduced  into  the  formula  in  the  man- 
ner just  indicated,  gives  the  truQ  value,  that  c  is  positive  and  less  than 
a  certain  quantity,  and  that  e  approaches  zero  if  h  approaches  zero.  We 
are  usually  not  able,  however,  to  specify  the  value  of  e  exactly ;  and  this 
is  not  necessary,  since  the  c  no  longer  occurs  in  the  equation  which  is 
deduced  from  that  in  h  by  equating  the  limits  of  both  its  members. 
It  is  sufficient  to  find  that  some  value  of  c  exists  such  that  for  it  the 
expression  in  question  gives  the  exact  change  which  takes  place  in  the 
interval  h,  and  second,  that  c  approaches  zero  when  h  does  so. 

The  following  presentation  will  illustrate  graphically  the  relation  of 
€  to  the  other  quantities.  Taking  the  data  from  the  above  problem 
(Newton's  law  of  cooling),  and  denoting  the  relation  between  the  tem- 
perature and  the  time  by  0  =f{t),  we  let  the  curve  CC  represent  the 
graph  of  k(0  -  Oq).  Let  OA  =t  a.nd  AB  =  h.  Then  A  C  =  k(0-  ^„) 
and  BD  =  k(0'  -  0^);  rectangle  ACFB  =  k(e  -  6(^)h,  and  rectangle 
AEDB  =  k(0'  —  O^h.  The  quantity  which  we  seek,  denoted  above  by 
W,  has  been  proved  to  be  larger  than  one 
of  these  rectangles  and  smaller  than  the 
other.  (Tt  will  be  shown  in  the  next  chap- 
ter that  W  is  precisely  the  area  A  CGDB.) 
If  we  move  the  line  ED  up  parallel  to 
itself,  there  must  be  some  position,  call  it 
MH,  such  that  rectangle  A  MHB  is  exactly 
equal  to  \V.  For  as  the  line  ED  moves 
up  parallel  to  itself,  the  rectangle  EDBA 
increases    continually,    and    passes    from 

being  smaller  than  W  (at  ED)  to  being  larger  than  W  (at  CF),  In 
doing  so  it  must  pass  through  a  position  in  which  the  rectangle  is  just 
equal  to  W.     The  distance  ME  is  the  graphic  equivalent  of  ke. 

We  may  also  show  graphically  that  c  is  less  than  $  —  0'.  For,  from 
the  values  of  ^C  and  BD  above,  we  have  CE  =  k(0  -  0').  But  ME 
=  ke,  and  ME <  CE;  consequently,  ke<ck($  —  0'),  or  €<cO  -  6'. 


230  CALCULUS  [Ch.  Vll. 

Art.  5.  Work  done  in  the  expansion  of  a  perfect  gas  at 
a  constant  temperature.  When  a  gas  kept  under  constant 
pressure  p  (the  temperature  also  being  constant)  expands 
by  the  volume  v,  it  is  known  that  the  work  done  is  equal  to 
the  product  pv,  but  if  the  temperature  alone  is  kept  con- 
stant, the  pressure  of  a  confined  mass  of  gas  will  change 
continually  as  the  volume  passes  from  the  value  v^  to  the 
value  v^. 

In  accordance  with  our  usual  method  of  procedure,  we 
assume  the  pressure  to  be  constant  during  an  expansion 
equal  to  h\  the  work  then  done  may  be  put  equal  to  ph. 
If  the  amount  of  work  done  in  the  expansion  of  the  gas 
from  the  initial  volume  v^  to  the  variable  volume  v  be 
denoted  by  L(^v}^  and  if  p  denote  the  pressure  when  the 
volume  is  v,  and  p'  that  when  the  volume  is  (v  +  ^),  then 
putting  0  <€<p  —  p',  we  have 

L(v  +  h)-  L(v}=(p-  e)h 


h 
As  h  approaches  zero,  this  approaches  the  limit 

dL 

dv 


or  — ^ f ^-^  =  p  —  e. 

lit 


(1)  ^^f 


and  by  integration, 

(2)  L=^pdv. 

It  is  to  be  observed  that  the  pressure  at  any  moment  is 
dependent  upon  the  volume,  or 

and  before  it  is  possible  to  carry  the  above  integration  any 
further,  the  nature  of  the  function  f(^)  must  be  known. 


5.]  APPLICATIONS  OF  INTEQRATIOH^  231 

We  assume  that  the  gas  obeys  Boyle's  Law,*  and  that  the 
expansion  occurs  at  uniform  temperature  ;  we  then  have 

pv  =  IC^ 

where  ^  is  a  constant  quantity  dependent  for  its  value  upon 
the  experimental  conditions  and  the  units  of  measurement. 
Consequently, 

and  by  substitution  in  equation  (2), 

J         V 

(3)  .  =  K\o^v-^r  C. 

When  v=  v^^  i  =  0 ;  for  if  there  were  no  expansion,  no 
work  would  be  performed.     Consequently, 

(4)  0  =  iTlog  v^  +  Q> 
Subtracting  (4)  from  (3),  we  have 

(5)  Z^iTlog-. 
Putting  V  =  v^^  we  have 

(6)  X=^l0g!2; 

but  v.  =  —  and  v^  =  — , 

so  that  by  substitution  in  (6) 

(7)  L=K\og^. 

*  This  assumption  is  quite  legitimate,  for  we  are  dealing  with  ideal  or 
perfect  gases,  one  of  the  definitions  of  which  is  that  they  are  such  gases  as 
are  strictly  subject  to  Boyle's  Law. 


232  CALCULUS  [Ch.  VII. 

The  last  equation  gives  the  work  done  during  the  expan- 
sion of  the  given  mass  of  gas  while  the  pressure  sinks  from^ 
the  value  p^  to  that  of  pg- 

Art.  6.  Work  done  in  the  expansion  of  a  highly  com- 
pressed gas  kept  at  constant  temperature.  If  a  given  mass 
of  gas  is  under  so  great  a  pressure  that  it  no  longer  obeys 
Boyle's  Law,  the  relationship  between  the  volume  and 
pressure  has  been  found  to  be  satisfactorily  given  by  van 
der  Waals's  equation  (p.  66): 

(IX  (^p  +  ^^(v-b}=K, 

where  a,  5,  and  K  are  constants  depending  upon  the  nature 
of  the  gas  and  the  conditions  and  units  of  measurement. 
Solving  this  equation  for  p,  we  have 

which,  when  substituted  in  equation  (2),  p.  230,  gives 

(4)  L  =  Klogiv  -b)  +  -  +  C, 
For  V  =  v^^  L  =  0^  so  that 

(5)  0  =  Klog(iv^-b}-{-~-hO. 

Subtracting  (5)  from  (4),  we  find 

(6)  x^^logii^-«(l-l\ 

or,  if  t>  =  Vg, 


5-7.]  APPLICATIONS   OF  INTEGRATION  233 

Art.  7.  Work  done  in  the  expansion  of  a  gas  undergoing 
dissociation  *  at  constant  temperature.  Consider  the  case  of 
a  gas  that  dissociates  on  expanding  so  that  some  of  its 
molecules  break  up  into  two  others  (binary  dissociation). 
If  the  gas  were  not  at  all  dissociated,  the  relation  between 
its  volume  v  and  pressure  p  is  given  (p.  3)  by 

(2)  pv  =  K, 

K  remaining  constant  during  the  expansion.  But  inasmuch 
as  the  gas  is  dissociated,  and  hence  contains  a  larger  number 
of  molecules,  the  pressure  is  greater,  increasing  with  the 
augmentation  of  the  number  of  molecules,  for  at  constant 
temperature  the  pressure  of  a  gas  is  directly  proportional  to 
the  number  of  molecules.  If  the  gas  at  first  contained  n 
undissociated  molecules,  and  if  the  fraction  x  is  dissociated, 
then  w(l  —  x)  represents  the  number  of  molecules  still 
undissociated,  and  2  nx  the  number  formed  by  dissociation. 
Hence  the  actual  pressure  P  is  to  the  pressure  p  with 
no  dissociation  as  the  number  of  molecules  present  in  the 
dissociated  gas  is  to  the  number  originally  present,  or 


P  :  p  =  n(l  —  x')-t  2nx  :  n  =  1  -\-  X  :  1, 


or 


(3)  P=(l+a;>. 


*  Many  gases  on  expanding  undergo  an  ever-increasing  dissociation ;  that 
is,  the  gaseous  molecule  breaks  up  into  two  or  more  constituent  molecules. 
When  the  gas  dissociates  into  two  products,  the  degree  of  dissociation  is 
equal  to  the  number  of  molecules  already  dissociated,  divided  by  the  number 
of  molecules  that  were  present  before  the  dissociation  began.  If  the  degree 
of  dissociation  be  denoted  by  x  and  the  volume  by  v,  the  equation 

(X)  ^^^ 

(k  being  a  constant)  has  been  shown  to  represent  the  facts  of  the  case. 


234  CALCULITS  CCh.  VII. 

We  found  above  (p.  230)  that  the  work  done  up  to  any 
instant  is  given  by 

(4)  L=fFdv, 

P  being  the  pressure  at  the  end  of  the  period  for  which  w^e 
compute  the  work.  In  the  case  in  hand  we  have  just  found 
that  pressure  to  be 

so  that 

(5)  Z  =  I  (1  +  x)p  dv  =  \  p  dv  -\-  \  px  dv, 

where  both  p  and  x  are  functions  of  v. 

The  first  integral  has  already  been  determined  (p.  231). 
It  remains  therefore  to  find 

(6)  j  px  dv. 
Expressing  everything  in  terms  of  x,  we  have 

(7)  I  pxdv  =  }  P^-T-  ^^' 

where  p  on  the  right  is  now  to  be  regarded  as  a  function  of  x. 
From  equation  (1),  in  footnote,  p.  233,  we  have 

dv  _  x(2  —  x^  ^ 
dx      k(l  —  0^)2 

From  (1)  and  (2),         p  =  ^^^C^"^). 

X 

We  have  then  to  find 

^  x^  k(\  —  xy 

or,  simplifying, 

^  \  —  X  ^ 

with  the  result, 

(8)  L  =  -K\og(l-x:)^Kx^-Q. 


7.]  APPLICATIONS   OF  INTEGRATION  235 

We  have,  therefore,  substituting  in  (5)  for  the  integrals 
their  values  from  p.  231  and  (8)  above,  respectively, 

(9)  L  =  K\og--K\\og(l-x)-x\  +  C. 

When  V  =  Vy,  i.e.  at  the  beginning  of  the  process,  no 
work  has  been  done,  since  there  has  not  yet  been  any  expan- 
sion. When  V  =^  v^^  X  will  have  the  value  x^,  which  can  be 
determined  from  equation  (1).     We  have  accordingly 

(10)  0  =  -  irsiog(i  -  x;)~x^\  +  a 

Subtracting  (10)  from  (9), 

(11)  L  =  K\\og^  +  x-x^-\ogl^:^^\' 

In  particular,  the  work  done  during  the  expansion  from 
the  volume  v-^  to  the  volume  v^  is 

(12)  L  =  ^|log^+  rr^  -x^-  log  f^^[. 

(  Vj  L  —  X^) 

As  remarked  above,  x^  and  x^  are  to  be  found  from  equa- 
tion (1) ;  their  values  are 


-'•^bi^-tr') 


The  formula  for  the  work  done  will  be  simpler  if  it  is 
expressed  in  terms  of  x-^  and  x^.     From  equation  (1), 

v^  =  1 and  ^2  = 


Kil-x{)  '      K(\-x^) 

and,  consequently,* 

*  Formulae  6  and  7,  Appendix. 


236  CALCULUS  [Ch.  vir. 

Art.  8.  Maximum  average  temperature  of  a  flame.  Be- 
fore taking  up  this  problem,  we  determine  the  amount  of 
heat  given  up  to  its  environment  by  a  body  of  mass  m 
when  its  temperature  falls  from  6^  to  6^  We  first  change 
the  meaning  of  W  from  that  which  it  had  on  p.  224.  There 
W(^0}  denoted  the  amount  of  heat  given  off  when  the  body 
cooled  from  the  initial  temperature  to  the  variable  tempera- 
ture 0 ;  here,  W(6}  shall  denote  the  amount  of  heat  given 
off  by  the  body  in  cooling  from  the  variable  temperature 
0  to  the  final  temperature  ^j,  the  temperature  of  its  environ- 
ment. This  is  necessary  since  the  initial  temperature  was 
known  in  the  previous  case,  while  here  it  is  the  quantity 
sought. 

If  the  specific  heat  e  (p.  226)  were  not  to  change  with  the 
temperature,  but  were  to  remain  constant,  the  quantity  of  heat 
TT  which  the  body  gives  off  in  cooling  would  have  the  value 

(1)  w=mcce,-0^-). 

But  this  condition  is  frequently  not  fulfilled,  as  specific  heat 
is  a  function  of  the  temperature.  Denoting  by  0^  a  constant 
temperature  arbitrarily  fixed  as  a  starting  point  for  the 
comparison  of  specific  heats  at  different  temperatures,  we 
may,  as  a  rule,  put 

(2)  c  =  a  +  ^{0 -  e^y  +  yio -  00^  +  8(0  -  e,y  +  -', 

where  a,  y8,  <y,  8,  •••,  are  constants  dependent  upon  the  par- 
ticular substances  under  consideration,  and  where  the  num- 
ber of  terms  of  the  expression  to  be  taken  is  regulated  by 
the  degree  of  accuracy  of  the  data.  The  significance  of  the 
expression  may  be  illustrated  by  observing  that  at  the 
temperature  Oq  all  the  terms  but  the  first  vanish,  so  that 
c  =  a.  When  the  temperature  of  the  body  falls  from  6  -\-  h 
to  ^,  the  specific  heat  decreases  from 


8]  APPLICATIONS   OF  INTEGRATION  237 

ci  =  a  -{-  jSiO -\-h  -  0^}  +  j(0  +  h -  e^y  +  s((9  +  A  -  e^y  +  ••  • 
to       c^  =  a  +  pie-  e^)  +  7  (^  -  ^o)'  +  ^  (^  -  ^o)'  +  -. 

If  the  specific  heat  had  the  value  c^  during  the  cooling 
through  the  temperature  interval  A,  the  amount  of  heat 
given  off  would  be  me^h^  while  if  the  specific  heat  were  c^^ 
the  heat  given  off  would  be  mc^h.  But  the  latter  amount 
is  too  small  and  the  former  too  large.  Hence  there  must 
be  some  intermediate  value  c'  such  that  mc'h  represents  the 
exact  amount  of  heat  given  off  in  the  temperature  interval. 
On  examining  the  form  of  c^  and  Cg,  we  see  that  the  con- 
dition 

c^<c'  <c^ 

is  satisfied  if  c^  has  the  form 

where  0<e<A.  We  have  then  as  the  true  amount  of  heat 
given  off 

(3)  mhia+fi^e+e-e.^+yie+e-e.y+Bce+e-e.y-h-i, 

Another  form  of  expressing  this  amount  of  heat  is 

(4)  wCO  +  h)-W(iO). 

Equating  (3)  and  (4)  and  dividing  through  by  A,  we  have 

(5)  Wje-hh^-WCO) 

h 

Letting  h  approach   the    limjt   zero,   and   noting   that    e 
approaches  zero  with  A,  we  have 

dW 


(6) 


de 
=  m\a  +  ^(e-o,)  +  y{e-  e,y  +  8(e-  e,y  + .  ••  j 


238  CALCULUS  [Ch.  VII. 

This  expression  could  be  integrated  directly  with  respect 
to  6  ;  but  it  will  be  more  convenient  later  on  to  deal  with 
an  integral  which,  like  the  above,  is  expressed  in  terms 
of  6  —  Oq;  we  introduce,  accordingly,  a  new  variable 
u=  0  —  6q,  integrate  with  respect  to  it,  and  then  replace 
u  again  by  its  value,  with  the  result : 

(7)  Tr=».{«(.-.„)  +  MzL^  +  Z(i^+...|+^. 

If  ^  =  ^j,  the  temperature  of  the  environment,  then  the 
amount  of  heat  given  off  in  cooling  to  that  temperature 
will  be  zero.      Consequently,  we  have 

(8)  0  =  .{<..-.„)+ ^(^^  +  ^^^I^V...}  +  C, 
or,  subtracting  (8)  from  (7), 

(9)  Tr=m{«(^-^i)  +  |[(^-^o)2-(^j-^,)^] 

+ 1  [(^-^o)'-(^i-^o)'] +  •••}• 


If  in  the  above  equation  we  put  6  =  6^^  we  evidently 
obtain  the  required  amount  of  heat  given  oft'  when  the 
body  cools  down  from  6^  to  6-^. 

The  following  simple  considerations  will  now  give  us  the 
maximum  average  temperature  of  a  flame.  The  elevation 
of  the  temperature  in  the  flame  is  occasioned  by  the  heat 
of  combustion  of  the  burning  substance.  The  heat  of  com- 
bustion must  be  equal  to  'the  amount  of  heat  which  the 
product  of  the  combustion  gives  out  in  cooling  from  the 
temperature  of  the  flame  to  that  of  the  chamber  in  which 
the  operation  takes  place  and  can  be  measured  by  conducting 
the  process  in  a  closed  and  isolated  vessel,  and  noting  the  rise 


8.]  APPLICATIONS  OF  INTEGRATION  239 

in  temperature.  We  know  then  the  amount  of  heat  that 
the  product  of  combustion  has  given  off  altogether,  and  we 
know  to  what  temperature  it  has  cooled  down,  viz.  the  final 
temperature  within  the  vessel,  and  we  seek  to  know  the 
temperature  of  the  product  when  the  cooling  process  began ; 
i.e.  in  equation  (9)  we  know  W  and  ^j,  and  we  seek  ^, 
which  can  accordingly  be  determined  from  this  equation, 
as  everything  else  in  it  is  known,  the  specific  heat  at  the 
temperature  0  being  as  above, 

c  =  a-{-^(e-  ^o)  +  7(^  -  o,y+ .... 

We  take  as  an  illustration  the  burning  of  carbon  monoxide 
in  pure  oxygen,  it  being  known  that  28  grams  of  carbon 
monoxide  unite  with  16  grams  of  oxygen  to  form  44  grams 
of  carbonic  acid  gas,  and  that  in  this  union  67,700  units  of 
heat  are  evolved.  The  specific  heat  of  the  carbonic  acid  gas 
referred  to  the  molecular  mass,  or  44,  as  unit,  has  been  found 
to  be  given  by  the  formula 

c?=  6.5 +  0.0084  (6>  +  273), 

where  a  =  6.5,  jS  =  0.0084,  and  0^  =  -  273°  C.  If  the  final 
temperature  in  the  chamber  be  ^^  =  0°  C,  the  substitution 
of  all  these  values  in  equation  (9)  gives 

67,700  =  6.5  ^  +  0:Mi£^i±l2i27M). 

The  solution  of  this  quadratic  equation  gives 

e  =  3205°  C. 

In  reality,  many  circumstances,  such  as  radiation,  dissocia- 
tion,*  etc.,  cause  the  actual  maximum  average  temperature 
to  be  less  than  this  theoretic  result. 

*  High  temperatures  as  well  as  other  reasons  cause  many  gases  to  break 
up  into  other  gases,  generally  of  simpler  nature, 

n 


240  CALCULUS  [Ch.  VII. 

Art.  9.  Chemical  reactions  in  which  the  factors  are 
totally  converted  into  products.  When  n  kinds  of  molecules 
enter  into  reaction,  the  speed  of  the  reaction  is,  according 
to  the  Law  of  Mass  Action  (p.  167),  proportional  to  the 
product  of  their  concentrations.*  To  simplify  matters,  we 
assume  that  equal  numbers  of  molecules  of  each  substance 
are  present,  and  that  the  concentration  of  each  may  be 
denoted  by  a  at  the  beginning  of  the  reaction.  Then, 
after  the  lapse  of  the  time  f,  the  concentration  of  each  will 
be  equal  to  (a  —  :r),  x  designating  the  amounts  of  the 
substances  chemicall}^  transformed. 

The  speed  of  reaction  is  therefore 

(1)  ^  =  k(a-xy, 

where  k  denotes  a  constant. 

The  integration,  with  respect  to  x,  of  the  reciprocal  of 
this  expression  gives 

(2)  —^ =jct-{-  a 


When  x=  0^  t  =  0,  so  that 
1 


^^^  T^TTTv:::^  -  ^' 


and,  by  subtraction. 

If,  for  example,  we  put  n  =  2,  then 

1  X 


(5)  k  = 


t  (a  —  x}a 


*  The  concentration  of  a  gas  may  be  defined  to  be  the  mass  contained  in 
the  unit  volume. 


9.]  APPLICATIONS   OF  INTEGRATION  241 

Equation  (4)  holds  only  when  n  >  1.  For  the  case  when 
n  =  1,  the  integration  leads  to  a  logarithmic  expression,  an 
example  of  which  we  have  already  met  in  the  inversion 
of  sugar  (p.  183). 

We  shall  now  proceed  to  consider  an  example  of  the  case 
where  the  initial  concentrations  of  the  substances  are  differ- 
ent. Let  two  different  kinds  of  molecules  react  upon  each 
other,  and  let  their  concentrations,  when  t  =  0^  he  a  and  b, 
respectively.  We  have  then,  at  the  time  t,  when  x  molecules 
of  each  substance  have  reacted,  the  speed  of  reaction  (cf .  (1) 
above), 
(6)  ^  =  k(:a-xXb-x-). 

Integrating  the  reciprocal  of  this  by  the  use  of  partial 
fractions,  we  find 

(7) ^{\og{b-x')-\og(a-x)']=J€t+  O, 

a  —  0 

When  X  =  0,  ^  =  0,  and 

0  = (log  b  -  log  a). 

a  —  0 

By  subtraction, 

^ox  1      1      (a  —  x)  b      J . 

(8)  — r  log^- ^-  =  kt. 

^^  a-b     ^(b-x)a 

If  in  equation  (8)  we  put  a  =  b^  equation  (5)  should 
result ;   but  we  encounter  here  the  peculiar  difficulty  that 

when  a  =  b^  the  first  factor  of  (8)  assumes  the  form  —  while 

the  logarithmic  expression  reduces  to  log  1 ;  that  is,  to  zero. 
This  difficulty  is  merely  apparent,  and  will  be  cleared  up  in 
Chap.  X. 


242  CALCULUS  [Ch.  VII. 

Art.  10.  Reactions  in  which  the  factors  are  only  partially 
converted  into  the  products.  In  a  reaction  occurring  in  a 
homogeneous  mixture  of  gases  or  a  homogeneous  solution 
at  constant  temperature,  the  Law  of  Mass  Action  states  the 
following  for  the  case  in  which  the  original  reactii^  sub- 
stances are  not  wholly  used  up  before  the  reaction  stops  : 
The  speed  of  reaction  at  any  moment  is  equal  to  the  product 
of  the  concentrations  of  the  reacting  substances  minus  the 
product  of  the  concentrations  of  the  substances  formed  by 
the  reaction,  each  product  being  multiplied  by  a  factor  of 
proportionality. 

Expressed  in  a  formula,  the  above  law  becomes 

(1)  p=k(ia-xXb-x^(e-x')"'-k'(a'  +  x}(b'+xXc'+x}"', 

where  x  represents  the  number  of  molecules  that  have  reacted 
in  the  time  ^,  and  a,  ^,  (?,•••  are  the  initial  concentrations 
(corresponding  to  the  time  t=  0),  a',  b'^  (?',•••  the  initial  con- 
centrations of  the  substances  formed,  and  k  and  k'  are  the 
constants  of  the  reaction. 

At  the  time  ^,  the  concentrations  of  the  reacting  sub- 
stances are  a  —  x^  b  —  x^  c  — rr,  •••,  while  those  of  the  sub- 
stances formed  in  the  reaction  are  a'  +  a;,  5'  +  a;,  c'  +  a^,  •-. 

Equation  (1)  is  integrable,  for  the  expression 

•    1 " 

k(a  -x}(b-  x}  ((?  -  x^ k^a'  +  x)(b'  +  x)  (c'  +  x)  - 

can  be  decomposed  into  partial  fractions  (p.  203).*  Such 
cases  as  have  as  yet  been  experimentally  studied  are  very 

*  The  integration  may  require  the  decomposition  of  the  denominator  into 
factors  of  the  first  or  the  second  degree.  While  this  is  theoretically  possible 
(i.e.  such  factors  exist),  it  may  not  always  be  possible  actually  to  find  them 
by  processes  of  algebra. 


10-11.]  APPLiCATtOJ^S  OP  tNTJEQBATION  248 

simple,  the  number  of  reacting  substances  never  exceeding 
three. 

Art.  11.  Formation  of  lactones.  We  take  up  one  example  illus- 
trating what  has  just  preceded.  Certain  organic  acids,  when  dissolved 
in  water,  form  compounds  known  as  lactones.  If  a  be  the  initial  con- 
centration of  the  acid,  and  a'  that  of  the  lactone,  then,  according  to 
equation  (1),  p.  242, 

(1)  y  =  k{a  -x)-  k'{a'  +  x), 

or,  taking  the  reciprocal,  and  integrating  with  respect  to  x, 

(2)  -1-  log  lijca  -  k'a'^  -  (k  +  k')x]  =  t  +  C. 
Since  t  and  x  vanish  together, 

(3)  ___L_log(^a-A:V)=C, 
and,  by  subtraction, 

(4)  log ^^  ~  ^'^' =(k  +  k')t. 

^  ^  ^  (ka  -  k'a')  -  (k  -f  k')x     ^  ^ 

At  the  expiration  of  a  very  long  time,  the  system  comes  into  a  state  of 
equilibrium ;  i.e.  no  further  reaction  takes  place.  In  that  case,  let  the 
concentrations  of  the  acid  and  lactone  be  A  and  A\  respectively.  Since, 
when  equilibrium  ensues,  the  speed  of  the  reaction  becomes  zero,  equa- 
tion (1)  assumes  the  form 

(5)  0  =  kA  -  k'A', 
or 

(6)  ]^^4!.  =  K, 
^  ^  k'      A  ' 

K  is  called  the  constant  of  equilibrium,  and  may  be  considered  to  be 
known  in  any  given  case,  since  it  can  be  found  experimentally.  By 
dividing  numerator  and  denominator  of  tlie  fraction  in  equation  (4) 
by  k',  we  obtain 

(7)  I  log Ka-a ^  ^      ^,,^ 

^  ^  t     ^{Ka-a')-{l  -V  K)x 

The  relationship  is  now  in  a  form  that  may  be  tested  by  experiment. 
Since  (k  +  k')  is  a  constant,  the  left  member  must  also  be  constant.  To 
give  a  numerical  example,  it  was  found  by  experiment  that  the  initial 


244  CALCtTLTTS  [Ch.  VIl 

concentration  of  acid  and  lactone  was  a  =  18.23  and  a'  =  0,  respectively, 
and  that  i<»^^ 

i^  =  ^  =  lM?  =  2.68. 
A       4.95 

For  corresponding  values  of  t  and  x,  the  following  values  for  equation 
(7)  were  computed : 

t  r  k  +  k'=-loo:.  ^^ 


21 

2.39 

36 

3.70 

50 

4.98 

65 

6.07 

80 

7.14 

120 

8.88 

160 

10.28 

220 

11.56 

320 

12.57 

t     "Ka-(1  +  K)x 

0.0350 
0.0355 
0.0370 
0.0392 
0.0392 
0.0375 
0.0376 
0.0371 
0.0357 


The  constancy  of  the  numbers  in  the  last  column  is  quite  satisfactory, 
and  indicates  that  the  assumptions  made  in  developing  the  theory  were 
correct  in  this  case. 


CHAPTER   VIII 


DEFINITE   INTEGRALS 


Art.  1.  The  quadrature*  of  the  parabola.  Let  it  be 
required  to  find  the  area  of  a  segment  of  a  parabola  which 
is  cut  off  by  a  straight  line  PP'  (Fig.  48)  perpendicular  to 
the  axis  of  the  parabola. 
The  axis  divides  this  seg- 
ment into  two  equal  parts, 
either  of  which  we  denote 
by  S.  If  we  draw  a  tan- 
gent to  the  parabola  at  its 
vertex  0,  and  let  fall  upon 
it  from  P  the  perpendic- 
ular PQ^  the  area  of  the 

segment  S  is  equal  to  the  difference  between  the  area  of  the 
rectangle  OQPL  and  that  of  the  figure  OPQ  bounded  by 
the  parabola  and  its  tangent.  We  need,  therefore,  to  deter- 
mine the  area  of  OPQ  only. 

Here  again  we  begin  with  a  method  of  approximations. 
We  take  the  tangent  at  the  vertex  of  the  parabola  as  the 
axis  of  abscissae,  and  divide  OQ^  which  we  denote  by  «,  into 
n  equal  parts  of  length  h ;  at  the  points  of  division  we  erect 
perpendiculars,  and  at  the  points  P^,  P^,  Pg,  •••,  where  they 
cut  the  parabola,  we  draw  lines  parallel  to  the  axis  of  x^  so 


*  To  find  the  area  of  a  curve  or  a  bounded  portion  of  a  plane  is  tanta- 
mount to  finding  the  area  of  an  equivalent  square. 

245 


246  CALCULUS  [Ch.  VIII. 

that  they  may  intersect  the  perpendiculars  at  the  points  i^^, 
i^j,  i^2'  ^3'  ••••     ^^  obtain  in  this  way  the  figure 

which  is  bounded  by  straight  lines,  and  contains  the  surface 
whose  area  we  have  to  find. 

We  can  determine  the  area  of  this  figure  as  follows  : 
Since  the  axis  of  ordinates  is  the  axis  of  symmetry  of  the 
parabola,  its  equation  reads  (p.  20), 

(1)  x^=2py,   or  y=^' 

If  ^1^1,  ^2^2'  ^3^3'  "*'  ^nVnip^n—  ^)  ^"^^  ^hc  coordluates  of 
the  points  P^  P^^  Pg,  •••,  P,  the  small  rectangles  lying 
between  two  successive  ordinates  have  areas  equal  to 

y^^  V'ifh  Vzh-,  •••,  ^A 
and  their  total  area  is  equal  to 

(2)  A  =  y^h  +  y^h  +  y^h-\-'-+yJi 

=  ^<^i  +  ^2  +  ^3+  •••  +^J- 
We  have 

(3)  x^=h,    x^  =  2h^   iTg  =  3  A,    •••,    Xn  =  nh  =  a^ 
and  according  to  (1), 

^^~2/   y^~    2p'   y^~    2p   '       '    ^"        2^ 
and  by  substitution  in  (2), 

(4)  A=—  (^2  +  22^2  +  32/^2  +  . . .  4-  ^2^2) 

2^ 

=  -^(1  +  22 +  32  +  .. -4-^2), 
2^ 


1.]  DEFINITE  INTEGRALS  247 


or. 


r.     A  ^  ^^  <n  +  1)(2  y^  +  1)  ^  JirKjhn  +  A) (2  hi  +.  h) 
^^  2p  1-2.3  1.2.3.2J9 

But  by  (3),  nh=  a;  hence, 


A  = 


a 


1  •  2  .  3  .  2  ;> 


(6)  =^  +  7,^  +  7,2     «. 

The  area  of  the  figure  bounded  by  straight  lines  approaches 
the  nearer  to  that  of  the  parabola,  the  smaller  the  quantity 
h  is  taken ;  that  is,  the  greater  the  number,  n,  of  the  rec- 
tangles, becomes.  Accordingly,  if  h  be  taken  small  enough 
the  area  of  the  figure  will  approach  just  as  nearly  as  we 
please  to  the  area  of  OFQ.  We  see  then  that  the  area  of 
OPQ  is  the  limit  which  the  area  of  the  polygon  approaches 
as  h  approaches  zero.  Denoting  the  area  of  OPQ  by  F^ 
we  have,  therefore. 


>r 

(8)  F="' 


The  area  of  the  segment  of  the  parabola  itself  may  now 
be  found  as  follows.  The  rectangle  OQPL  has  the  sides 
OQ=a  and  PQ^y^.  Its  area  is  hence  ay^^  and  may  be 
expressed  in  terms  of  a  by  equation  (1)  with  the  result 

"^^^  =  2^* 
*  Formula  53,  Appendix. 


248  CALCULUS  [Ch.  VIII. 

The  area  of  the  segment  of  the  parabola,  S^  that  is  bounded 
by  PL  and  the  axis  of  ordinates  of  the  parabola,  is 

(9)  S^^-^  =  ±; 

Ip      bp      3jt? 

therefore, 

(10)  jS=2F, 

Accordingly,  the  parabola  divides  the  rectangle  into  two 
parts^  one  of  which  is  twice  the  other. 

In  the  problem  just  solved  we  find  again  confirmed  that 
which  was  said,  p.  101.  Even  at  those  decisive  points  where 
our  conceptions  lose  their  definiteness  and  become  obscure, 
our  calculations  lose  nothing  in  definiteness  and  clearness; 
they  furnish  what  our  conceptions  are  unable  to  furnish. 
An  analogous  state  of  affairs  occurs  in  every  case  where 
we  have  to  determine  a  sum,  the  number  of  whose  parts 
is  increasing  without  bound  while  the  magnitude  of  each 
separate  part  approaches  zero.  The  area  of  a  surface  of 
any  shape,  the  contents  of  a  body  with  variable  cross-section, 
the  total  mass  of  a  body  of  varying  density,  the  sum  of 
all  the  attracting  forces  that  are  exerted  on  a  point  by  all 
the  parts  of  a  body, — all  these  are  examples  of  the  class 
of  problems  that  may  be  handled  in  the  way  just  set  forth. 

Art.  2.   Notation  of  sums.     For  the  sum 

occurring  in  equation  (2),  p.  246,  the  abbreviation 

S(^A)  or  %7/h 

has   been  adopted,  where  S  indicates  that  a   sum  is  to  be 
formed  of  terms  like  i/h,  in  which  all  the  values  of  ?/,  as 


1-3.]  DEFINITE  INTEGRALS  249 

^v  Vv  ^3'  •"'  ^«'  ^^"®  substituted  in  order.  Another  cus- 
tomary notation  is  to  write  t^x  instead  of  h,  to  bring  out 
the  fact  that  Ji  is  the  difference  of  the  abscissae  for  two 
successive  points  of  division.     We  have,  accordingly, 

(1)  A  =  ^yh  and  A  =  lyAx, 

as  expressions  for  the  sum  of  all  the  rectangles.  The  limit 
of  either,  when  h  (or  Ax^  approaches  the  limit  zero,  is  the 
area,  F,  of  the  segment  of  the  parabola,  i.e. 

(2)  F  =  ."f'o 2^^'  o-^  ^  =  Ax" 0  [^3/A*] . 

We   shall   now  prove   that   this   limiting  value   may  be 
found  by  substituting  x  =  a  in  the  value  of 


(3)  J^/A 


(or,  speaking  more  exactly,  in  one  of  the  boundless  number 
of  values  which,  as  we  know,  this  integral  has).  For  if  we 
substitute  for  y  in  (3)  its  value  given  in  (1),  p.  246,  we  find 

(4)  fi,dx  =  f^dx  =  ^^fx^dx 

and  if  we  take  the  particular  case  of  this  in  which  (7=  0, 
and  in  it  put  rr  =  a,  we  get  the  same  expression  that  we 
found  in  (8),  p.  247,  for  the  area  F. 

Art.  3.    The  quadrature  of  any  curve.     What  precedes 
can  readily  be  extended  to  any  curve  whatever.     Let 

(1)  y=/(a=) 

be  the  equation  of  the  curve.  We  assume  that  the  curve 
intersects  the  axis  of  ordinates,  as  is  actually  the  case  in 


250 


CALCULUS 


[ch.  vm. 


Fig.  49,  and  proceed  to  calculate  the  area  of  the  surface 
bounded  by  the  axis  of  rr,   that  of  y,  the  curve,  and  the 

ordinate  PQ.  As  in  the  previous 
example,  we  imagine  the  abscissa  OQ 
(which  we  denote  by  a)  to  be  di- 
vided up  into  any  number  of  small 
parts,  which  are  not  necessarily 
equal,  and  denote  them  by.  A^,  h^^ 
A3, .-..  We  then  conceive  of  perpen- 
diculars being  erected  at  the  points 
of  division  and  cutting  the  curve  at 
and  consider  the  figure  bounded  by  the 
Its  area  approximates  to  that  of  the  given 


Fig.  4i>. 


-t  J,    X^2'     ^ 


straight  lines. 

surface.  Although  a  portion  of  this  figure  projects  above 
the  curve  and  a  portion  lies  below  it,  that  is  of  no  conse- 
quence in  our  results.  If  the  coordinates  of  P^,  P^,  Pg,  ••-, 
be  denoted  by 

^iVv     ^iVt'     ^3^3'     ***' 

the  area  A  of  this  surface  is 

(2)  A  =  \y^  +  %2  +  %3  ••• ; 
or,  if  we  put 

(3)  Aj  =  rr^  —  cPq  =  AiTj,    li^  —  x^  —  x^  =  Ax2  •••, 

then  A  =  y^Ax^-{-  7/2^^2  ~^  '"y 

and,  on  using  the  notation  of  sums, 

(4)  A  =  ^yh  =  l^yAx. 

We  know  that  this  sum  still  represents  the  area  of  the 
surface  when  all  the  A's  or  the  A^^'s  approach  zero ;  the  sum 
has,  therefore,  a  definite  limiting  value  when  h  approaches 
zero,  viz.  the  area  of  the  surface  P^OQP;  and  we  have, 
letting  lim  A  =  F(a)^ 


3]  DEFINITE  INTEGRALS  251 

(5)  F(a)=\iml^y^x']. 

We  shall  prove  that  this  limit  is  also  represented  by  one 
of  the  values  of  the  integral 

(6)  jy  dx=:jf(x)dx 

when    in    it    we    put   x  =  a.      To    do   this   we    must   show 
that   F(x)    is    one    of   the   values    of    \f(x)dx',    i.e.    that 

In  order  to  obtain  the  derivative  of  F(x),  we  have  to  find 

the  limit  of 

F(x  +  h)-  F(x) 

1 ' 

as  h  approaches  zero. 

To  this  end,  we  change  the  notation  slightly,  and  let  the 
l)()int  Q  now  be  a  variable  point,  and  denote  the  distance 
OQ  hj  X.  We  also  designate  by  P'  the  point  in  the  curve 
that  corresponds  to  the  abscissa  x  +  h  =  OQ'^  ^o  that 

Fix  +  li)=OP^P'Q', 

F(x)^OP,PQ, 

and,  by  subtraction, 

F{x  +  K)-  F(x)  =  PQP'Q\ 

There  must  be  between  P  and  P'  a  point  with  the  coor- 
dinates f  and  77,  such  that  the  surface  PQP'Q'  will  be 
equal  to  a  rectangle  with  the  base  QQ'  =  h  and  the  altitude 
^7  =/(?);  that  is, 

,     F(x  +  h^-F(^x^=hrj  =  hfQy, 
hence 

(7)  Fjx^-h^-Fjx^^j,^. 

h 


252  CALCULUS  [Ch.  VIII. 

If  we  now  let  h  approach  zero,  P'  (as  well  as  the  point 
lying  between  P^  and  P,  and  having  the  coordinates  f  and 
77)  will  approach  the  limit  P ;  that  is  to  say,  |  will  approach 
x^  and  hence 
r^\  lin^  F(x-{-h)-Fx  _  ...     . 

or, 

(9)  •  ^  =/(-)' 
hence, 

(10)  FQx^  =  J/(^)  dx, 

which  was  to  be  shown. 

Combining  (5)  and  (10),  and  remembering  that  the  con- 
stant abscissa,  a,  has  been  replaced  above  by  the  variable 
abscissa,  x^  we  find 

(11)  F(x)==jy  dx  =  lim  \ly  H^x], 

and  this  equation  states  that  the  integral  of  the  func- 
tion y  is  nothing  other  than  the  limit  which  this  sum 
approaches  when  the  parts  h  or  Aa;  into  which  the  a:-axis  is 
divided  approach  the  limit  zero.  From  this  fact  the  nota- 
tion for  integrals  arose.  The  sign  j  ,  proposed  by  Leibnitz 
in  the  early  development  of  our  subject,  represents  a  form 
of  s  now  obsolete,  standing  for  the  word  sum^  and  y  dx  repre- 
sents the  type  of  the  terms  of  the  sum.  The  portion  dx  of 
the  symbol  indicates  which  variable  it  is  whose  increment  h 
or  A2:  is  made  to  approach  zero  to  obtain  the  limit  in  ques- 
tion. This  variable  is,  of  course,  that  with  respect  to  which 
under  the  other  definition  of  integral  we  should  have  to 
differentiate  the  integral  in  order  to  obtain  the  function 
under  the  sign. 


3-4.]  DEFINITE  INTEGRALS  253 

To  reconcile  this  geometric  interpretation  of  the  integral 
with  its  boundless  number  of  values,  we  consider  that,  while 
an  integral  represents  an  area  one  of  whose  boundaries  is 
the  axis  of  ordinates,  we  are  quite  free  in  our  choice  of  the 
position  of  this  axis.  Hence  it  is  easily  seen  that  there  can 
be  an  indefinite  number  of  values  for  each  integral,  and 
furthermore,  that  two  of  these  valvies  (which  are  functions 
of  x)  can,  for  equal  values  of  x^  differ  only  by  a  constant 
equal  to  the  area  comprised  between  the  two  axes  of  ordi- 
nates under  consideration. 

Art.  4.  Definite  integrals.  We  now  propose  to  deter- 
mine the  area  of  a  figure  lying  between  an^/  two  ordinates 
of  a  curve,  as  F^Q^=b-^^  and  P^Q^=h^  (I^'ig-  ^^^  P-  '^^^^' 
These  ordinates  may  have  any  position  whatever,  and  we 
denote  their  abscissae  by  a^  and  a.^  The  surface  F  which 
they  bound  is  the  difference  between  surfaces  bounded  on 
the  one  side  by  the  axis  of  ordinates,  and  on  the  other  by 
P2Q2,  ^^^  P\Qv  respectively.  Their  values  are  F(^a^  and 
F(a^^  so  that 

(1)  F=Fia,)-F(ia,-). 

For  this  case  the  notation 

(2)  F^pydx 

has  been  adopted,  meaning  that  the  right  member  of  the 
equation  is  equal  to  the  difference  F(a^  —  F(^a^)  ;  that  is, 

(^>  j^_VW<^a.  =  J'(a,)-J'(aO. 

Such  an  integral  is  termed  a  definite  integral ;  a^  is  called 
its  lower  limit  and  a^  its  upper  limit. 

In  order  actually  to  find  the  value  of  the  definite  integral 


254  CALCULUS  [Ch.  VIII. 

as  here  defined,  it  would  be  necessary  to  find  F^x)  such 
that  its  derivative  is  f(x).  (If  the  function  f{x)  is  at  all 
complicated,  this  may  be  beyond  our  skill.  Still  it  is  often 
possible  to  find  the  value  of  the  definite  integral,  even  when 
we  are  unable  to  determine  the  function  F(^x)'). 

As  illustration,  we  take  the  case  of  the  parabola  for  which 
we  have  seen  (p.  247)  that 

then,  if  a^  and  a^  be  the  abscissae  of  the  points  P^  and 
^21  respectively, 

•^"i  h  p      bp 

Since  a  definite  integral  gives  the  area  of  a  surface  with 
definite  boundaries,  its  value  must  of  course  be  a  defifiite 
number.  It  must  be  independent  of  the  value  of  the  con- 
stant of  integration ;  that  is,  it  does  not  depend  upon  the 
position  of  the  axis  of  ordinates.  As  a  matter  of  fact,  it 
appears  clearly  from  the  above  discussion  that  whatever  the 
constant  is,  it  is  both  added  and  subtracted  in  forming  the 
definite  integral,  and  therefore  disappears. 

In  contradistinction  to  the  definite  integral,  the  function 
F(x}  is  called  an  indefinite  integral,  and  the  following 
notation  is  also  sometimes  used : 


I        7/dx  = 

a. 


F(x), 


the  right  member  as  well  as  the  left  being  equivalent  to  the 
difference  ^(^2)— ^(<^i)- 

In  words :  The  definite  integral  is  equal  to  the  difference 
between  the  values  of  the  indefinite  integral  for  the  upper 
and  the  lower  limit  of  integration. 


4-5.] 


DEFINITE  INTEGRALS 


255 


A  similar  notation  is  customary  when  the  symbol  /  is 
used  to  express  the  sum  of  the  products  corresponding  to  a 
division  of  a  fixed  portion  of  the  a^-axis  into  parts.  Hitherto 
we  have  either  regarded  the  boundaries  of  the  portion  of  the 
axis  which  was  divided  as  understood,  or  we  have  specified 
them  in  words.     They  may  be  more  conveniently  indicated 

(read  "  sum  from  a^  to  a^  "),  a^  being  the  abscissa 

of  the   end  where  the  summation   begins,   and   a^  of   that 
where  it  ends. 

Art.  5.  The  quadrature  of  the  ellipse  and  of  the  hyper- 
bola. Assuming  the  axes  in  their  customary  position,  the 
equation  of  the  ellipse  is 

(1)  ~ 


^  -4-  '^  =  1 

a?      IP' 


The  area  of  the  surface  Pj^^  ^2^2 
(Fig.  50)  is 

(2)        P,Q,Q.,P^=Cydx. 

From  the  equation  of  the  ellipse, 


(3) 


y 


V-- 


x' 


and  by  substituting  this  value   in   the  integral,   and  inte- 
grating, we  obtain 

P^Q^Q^P^  —  —    ^ix^a?  —  x^  +  a^  arc  sin  -  ]. 

--♦ 

Putting  ^2  =  a,    a^  =  0,  we  have   — -—   as  the  area  of  a 

quadrant  of  the  ellipse. 
18 


256 


CALCULUS 


[Ch.  VIII. 


This  result  can  also  be  obtained  otherwise.  We  first 
show  that  the  coordinates  of  any  point  of  the  ellipse  have 
the  values 

(4)  x=^a  cos  <^, 

y  =  h  sin  <^, 

in  which  the  angle  <^  is  defined  as  follows:  Constructing 
the  auxiliary  circle  (p.  51)  of  the  ellipse,  and  prolonging  PQ 
to  P'  (1^  ig-  51),  we  denote  the  angle 
F'OQhj  <l>.*     In  the  triangle  F'  OQ, 

X  =  a  cos</), 

and  if  this  be  substituted  in  equation 
(3),  it  is  seen  that 

?/  =  Jsin<^.f 

We  have  now  to  determine  the  value 
of  the  integral  (2),  regarded  as  an  indefinite  integral.  By 
differentiating  equation  (4),  we  find 


Fig.  51. 


and 


^  =  -asin<^, 
d(p 


(5)    \  ydx  =  j  y--—d(j)  =  —  \  ah  sin^ (f)d<f)  =  —  ab  \  sm^(f>d(l> 


d<f> 


^J(l- cos  2  (/>)#$ 


ah  ,    ,  ab  .    i^  , 

The  value  of  the  definite  integral  is  to  be  found  by  intro- 
ducing the  limits,  which  were  originally  a.^  and  a^,  but  we 


*  The  angle  P'OQ  is  called  the  eccentric  angle  of  the  point  P. 
t  Formula  29,  Appendix.         \  Formula  37,  Appendix. 


5.J  DEFINITE  INTEGRALS  257 

have  since  introduced  a  new  variable  ^,  and  accordingly 
have  </)2  and  (/)j  as  the  values  of  </>  corresponding  to  the 
points  Py  and  F^  The  area  E  of  the  segment  P^Q^Q^P^ 
is  therefore 


(6)  E 


ah  ,    ^   ah    .    ^ 


=  ^  (^1  -  ^2)  -  X  (si"  2  </>!  -  sin  2  (/>2). 


In  particular,  if  P^  coincides  with  A^  and  P^  with  j5j, 
hence,  denoting  the  area  of  a  quadrant  by  U^^ 


(7)  U^ 


2      2       4 


TT. 


Accordingly^  the  area  of  the  ivhole  ellipse  is  equal  to  ahir. 

This  formula  is  closely  related  to  the  formula  for  the  area 
of  a  circle.  The  area  of  the  auxiliary  circle  is  a'^ir ;  the 
area  of  the  ellipse  is  derived  from  it  by  putting  the  minor 
semi-axis  h  instead  of  one  of  the  factors  a.  This  is  in 
agreement  with  the  fact  that  the  ratio  of  any  ordinate 
of  the  ellipse  to  that  ordinate  of  the  circle  which  has  the 
same  abscissa  is  h  :  a.     (See  Eq.  3,  p.  51.) 

We  shall  carry  out  the  quadrature  of  the  hyperbola  only 
for  the  particular  case  that  the  hyperbola  is  equilateral ;  its 
equation,  referred  to  its  asymptotes  as  axes,  is  (p.  60) 

(8)  xy  =  a,  or  y  =  ^. 


258 


CALCULUS 


[Ch.  VIII. 


Therefore,  the  area  H  of  the  portion  of  its  surface,  bounded 
by  the  two  ordinates  y^  and  y^  whose  abscissBe  are  x^  and  x^^  is 


H 


?/  aa;  =  I     -dx  = 


^2 


logo; 


Fig.  52. 


and     (9)  H=a(\ogx^-\ogx^)=^  alog^- 

The  area  of  any  portion  of  the 
surface  of  an  hyperbola  is  repre- 
sented by  this  simple  formula. 
If  we  take  the  lower  limit  at  a 
point  whose  abscissa  is  ^i  =  l,  we 
find,  on  denoting  the  upper  limit 
hj  x, 
(12)  H=a\ogx 

as  the  area  in  question.     On  ac- 
count  of    this   relation,   natural 
logarithms  are  sometimes  very  appropriately  termed  hyper- 
bolic logarithms. 

Art.  6.  The  volume  of  a  solid.  In  order  to  calculate 
the  volume  of  a  solid,  —  that  of  the  sphere,  for  instance, 
—  the  solid  is  conceived  to  be  divided  up  by  parallel  planes 
into  constituent  parts,  just  as  a  surface  was  divided  up  by 
ordinates  into  constituent  areas.  A  right  cylinder  (usually 
of  irregular  base)  can  be  substituted  for  such  a  constituent 
solid,  just  as  a  rectangle  was  put  for  a  constituent  area ; 
then  the  limit  of  the  sum  of  all  such  cylinders,  when  their 
altitudes  are  made  to  approach  zero,  is  the  volume  of  the 
solid.  This  corresponds  to  the  way  in  which  geographic 
relief  maps  of  great  accuracy  can  be  prepared  by  the  super- 
position of  properly  cut  sheets  of  paper. 


5-70 


DEFINITE  INTEGRALS 


259 


We  designate  the  volume  of  the  solid  by  FJ  and  divide  the 
altitude  H  into  a  number  of  equal  parts,  each  of  magnitude 
h  ;  through  the  points  of  division  we 
pass  parallel  planes  ;  let  g  denote  the 
area  of  the  cross-section  of  the  solid 
made  by  one  of  these  planes,  then  gh 
will  be  the  volume  of  the  right  cylin- 
der of  altitude  ^,  on  ^  as  a  base  ;  the 
smaller  h  is,  the  less  the  cylinder  differs 
from  the  segment  of  the  solid  included 
between  the  same  planes.  The  volume 
of  the  solid  is  the  limit  which  the  sum  of  all  .the  cylinders 
approaches,  as  the  number  of  parts  into  which  the  altitude 
is  divided  is  made  large  without  limit,  and  consequently 
each  part  (denoted  above  by  A)  approaches  zero. 

We  have  then 

lim 


Fig.  53. 


v= 


tgh. 


or 


V=jgdH, 


taken  between  limits  corresponding  to  the  two  end  points 
of  the  altitude  II\  dH  is  a  part  of  the  integral  symbol, 
because  h  is  the  increment  of  the  altitude  H. 


Art.  7.  The  volume  of  the  sphere  and  of  the  paraboloid 
of  revolution.  In  the  case  of  the  sphere,  the  base  is  a  small 
circle  formed  by  an  intersecting  plane  at  the  distance  H 
from  the  center,  and  is  equal  to 

(1)  g  =  pV, 

p  being  the  radius  of  the  small  circle  (Fig.  53). 
But  if  r  be  the  radius  of  the  sphere, 

(2)  p^  =  r^-H\  2im\g  =  {r'^-H^)Tr, 


260  CALCULlfS  [Ch.  VIII. 

whence  the  volume  of  the  cylinder  at  that  point  is 

(3)  V  =  ir(r^  -  H'^^h, 
and  the  volume  of  the  sphere  is 

(4)  V =Xr^^^  ~  ^'^^'^  ^^' 
By  indefinite  integration, 

and  on  substituting  for  H  the  upper  limit  r  and  the  lower 
limit  —  r,  the  volume  K  of  the  sphere  is 

(5)  •      K=  ''  irfrW- 

='i(-i)H--j)i' 

(6)  -if^- 

The  volume  of  any  solid  of  revolution  may  be  found  in 
a  similar  way.  We  determine  as  further  illustration  the 
volume  of  a  solid  that  is  bounded  by  a  surface  generated 
by  the  revolution  of  a  parabola  around  its  axis.  This  sur- 
face is  called  the  paraboloid  of  revolution.  Let  the  equation 
of  the  parabola  be  x^  =  2  fy^  and  let  the  altitude  of  the 
parabolic  segment  be  H.  We  consider  the  volume  to  be 
the  limit  of  the  sum  of  circular  cylinders  at  distance  y  from 
the  vertex  of  the  parabola,  the  cylinder  being,  accordingly, 
of  radius  x^  and  its  thickness  Ay  approaching  zero  as  a  limit. 
We  have  then  for  the  volume  of  one  of  the  cylinders, 

(T)  V^  =  irx^Ay, 

or  by  substituting  the  value  of  x\ 

V  =  2  irpyAy, 


7-8.]  DEFINITE  INTEGRALS  261 

\yheiice 


(8)  V=j\pirydy  = 


H 

y'^pir, 

0 


or     (9)  V=H'^pir. 

The  formula  shows  that  the  volume  of  a  paraboloid  of 
revolution  is  equal  to  that  of  a  right  cylinder  with  radius  H 
and  altitude  p.  All  of  its  segments  can,  accordingly,  be 
represented  by  cylinders  with  a  constant  altitude,  but  with 
a  variable  radius  y. 

Art.  8.  The  mass  of  a  rod  of  varying  density.  To  deter- 
mine the  mass  of  a  right  cylindric  rod  whose  density  varies 
as  the  cube  of  the  distance  from  one  end. 

Let  L  be  the  length  of  the  rod,  and  a  the  area  of  its  cross- 
section.     We  divide  the  rod  into  n  equal  parts,  the  length 

of  each  being,  accordingly,  — ,  and  its  volume The  hth. 

n  n 

part  has  its  nearer  end  at  the  distance —,,  and  its 

kL  ^ 

farther  end  at  the  distance  —  from  the  end  from  which 

n 

we  measure.     The  densities  at  the  two  ends  of  these  parts 

are  then  c  \  ^ ^  >    and  c    —  >  ,  respectively,  c  being  a 

(         n        )  K  n  \ 

constant. 

The  mass  of  this  part,  (mass  equals  density  times  volume,) 

is  then  greater  than  ca(h —\y'[  —  \  and  less  than  cak^ 

The  mass  of  the  whole  rod  is  greater  than 


(1)  Z'^''<*-^>'SJ 


7  caKjz  —  iy\ 
and  less  than 
(2)  I-<f 


262  CALCULUS  [Ch.  Vlll. 

Taking  out  the  constant  factors,  we  may  write  these  sums 

where  Ic  assumes  the  values  1,  2,  3,  4---7i  (since  we  have  the 
sum  of  all  the  n  parts).     If  the  number  of  parts  is  increased 
without  bound,  the  limit  which  either  of  the  above  expres- 
sions approaches  is  the  mass  sought. 
We  may  write 


(*>     i^^  "•  I 


n     J       71 


takes  the  values  0,  -,    -, ; 

n  n     n     n         n 

i.e.  we  have  the  interval  from  zero  to  unity  divided  into  n 
equal  parts.     Putting  as  usual,  -  =  A,  we  have  to  determine 

But  this  is,  by  the  definition  of  definite  integrals, 
(6)  £a^dx. 

The  indefinite  integral  is  —,  and  the  value  of  the  definite 
integral  is,  accordingly,  ^. 

The  mass  of  the  rod,  which  was  found  to  be  caL^  times 
the  limit  of  S,  is  therefore 

4    • 

Art.  9.  Some  laws  of  operation  for  definite  integrals. 
Since  definite  integrals  are  defined  as  the  limiting  values  of 


8-9.]  DEFINITE  INTEGRALS  '26^ 

sums,  and  may  represent  areas,  volumes,  etc.,  they  permit  of 
the  application  of  certain  laws  of  operation.  An  area  may 
be  divided  up  into  parts,  and  the  problem  of  finding  its 
value  may  be  solved  by  determining  the  area  of  each  of  its 
parts.  Likewise,  the  calculation  of  a  sum  can  be  reduced 
to  the  calculation  of  the  parts  into  which  the  whole  sum  is 
divided.  From  this  self-evident  principle  several  rules  of 
operation  are  readily  deduced. 

If  the  limits  of  one  integral  are  a  and  5,  and  those  of  another 
integral  of  the  same  function  are  h  and  6',  then 

(1)  Cf{x)  dx  +  Cfix-)  dx  =  Cf(x)  dx ; 

this  means  simply  that  the  area  from  a  to  c  is  equal  to  the 
sum  of  the  areas  from  a  to  h  and  from  h  to  c.  A  similar 
statement  is  true  of  a  sum  of  more  than  two  of  such 
integrals. 

A  second  rule  is  the  following.     According  to  the  defini- 
tion of  a  definite  integral, 

f(x)  dx  =  lim  '^f(x)  Ax, 

all  of  the  quantities  Ax  filling  up  together  the  distance 
between  the  abscissae  a^  and  a^,  so  that  their  sum  is  equal 
to  ^2  ~  ^1*     Similarly,  the  definite  integral, 


f(x)  dx, 


having  a^  for  its  upper  and  a^  for  its  lower  limit,  is  equal  to 
such  a  limiting  value,  with  this  difference,  however,  the  sum 
of  all  the  quantities  Ax  must  now  be  equal  to  a^  —  a^^  they 
being  in  this  case  taken  with  signs  opposite  to  those  of 
the  first  integral.   'We  have  then 


264  CALCULUS  fCH.  VIII. 

(2)  Cfix^dx^-rfix-ydx. 

In  words :  A  definite  integral  changes  sign  when  its  limits 
are  interchanged, 

.     This  may  also  be  shown  in  another  way.     We  have  found 
that 

rfiix)dx  =  Fia^^-Fia{), 

where  F(x)  is  a  function  such  that  — r^-^=/(^);   accord- 

,  dx 

ingly, 

Cfix)  dx  =  FCa,)  -  Fia,-)  =  -  Cfix)  dx. 

This  law  is  a  special  case  of  the  general  mathematical 
principle  that  the  opposition  of  positive  to  negative  can  be 
geometrically  expressed  by  an  opposition  of  direction  ;  in 
this  case  the  direction  of  integration^  by  which  term  we 
understand  the  direction  in  which  the  independent  variable 
X  increases. 

In  equation  (1)  the  condition  was  implied  that  the  abscis- 
sae a,  5,  c  were  in  the  order  of  increasing  magnitude.  The 
equation  is  correct,  however,  even  though  this  be  not  the 
case.  If,  for  example,  a  <  c  <b^  we  have,  in  accordance 
with  equation  (1), 

r/(^)  dx  +  f /(:r)  dx  =  f /(2:)  dx, 

•/a  »/c  c/a 

f(x)  dx  =  —  i   f(x')  dx  ; 

c  Jb 

if  we  now  subtract  this  equation  from  the  preceding  one, 
the  remainder  is  ' 


9.]  DEFINITE  INTEGRALS  265 

(3)  f /(a^)  dx=C  'fix)  dx  +  ffCx-)  dx ; 

%Ja  A/a  »/6 

therefore  equation  (1)  is  true  even  for  this  case.*  / 

EXERCISES    XXVI 

Find  the  value  of  the  following  definite  integrals  \/ 

J  a  4  J2a    3;-2^>-     \ 


2.   J^   x'^dx. 

Ans.  219. 

11. 

3.   J    co^xdx. 

^n5.  1. 

12. 

n      xdx 

^\  VI  -  x^ 

4.  j-;..^x. 

Ans.  e«  — e^ 

13. 

p        ./x 

Ans.  ^. 

J"    Va--^-:.-^ 

6 

5.  (  ''cos  X  cfx. 

6.  (e-'^dx. 

Ans.  0. 

Ans.  1  -i. 
e 

14. 
15. 

Ans.  e  —  \. 

'•  ri- 

Ans.  1. 

16. 

Ans.  a. 

a  f'^ 

17. 

Ans.  n  —  r. 

9.   j'^'a,-9(fa. 

18. 

\      sm  X  dx. 

Ans.  0. 

*  We  have  tacitly  supposed  throughout  that  the  function  to  be  integrated  ( 

does  not  beco^ie  infinite  for  any  value  of  x  between  the  limits  of  integration, 
inclusive ;  (geometrically,  that  the  curve  whose  area  we  find  has  no  infinite  / 
branch  between  the  limits.)  In  case  the  function  becomes  infinite  for  ^ 
values  of  x  between  the  limits,  our  results  do  not  necessarily  hold,  but 
require  further  investigation,  to  make  which  would  be  beyond  the  scope  of 
this  work.  We  therefore  presuppose  in  every  case  that  the  function  to  be 
integrated  does  not  become  infinite,  or  otherwise  discontinuous,  between  the 
limits. 


%6                                              CALCULUS  '     [Ch.  VIII. 

19.  (       cos^xdx.                 Ans.   -•       21.    (    tanxclx.  Ans.  -^^• 

20.  f  ^sin^xdx.                 Ans.  '^'       22.    T      "^^    ♦  ^ws.  loff2. 
23.    i (put  c*  =  m).                                         ^ns.  arc  tan  e 


4 
24.    1    co^^xdx.  Ans.  |. 

25.  The  equation  of  the  equilateral  hypeibola  referred  to  its  asymp- 
totes as  axes  is  (p.  61) 

xy  —  a^. 

Find  the  area  included  between  the  curve,  the  axes  of  x,  and  the 
ordinates  a:  =  a  and  x  =  2a.  Ans.  a^log2. 

26.  Find  the  area  included  between  the  curve  ij  =  5x*  and  the  x-axis 
from  the  origin  to  the  ordinate  x  =  10.  Ans.  100,000. 

27.  Find  the  area  between  the  curve  y  =  e*,  the  axis  of  x,  and  the 
ordinates  x  =  1  and  x  =  2.  Ans.  e{e  —  1). 

28.  Show  that  the  area  of  the  segment  of  the  hyperbola 
cut  off  by  the  ordinate  at  a:  =  c,  is 


J  -  Vc'^  —  cfi  —  a  log 
(  a 


29.  Find  the  mass  of  a  right  cylindrical  rod  in  which  the  density 

caL" 
varies  as  the  distance  from  one  end.  Ans. 

30.  Find  the  mass  of  a  similar  rod  when  the  density  varies  as  the 
seventh  power  of  the  distance  from  one  end. 

31.  Show  that  the  volume  (oblate  spheroid)  generated  by  revolving 
the  ellipse 


^4.^  =  1 
a^      62 


•     .    4tTra% 
about  its  minor  axis,  is  — - — 


9-10.] 


DEFINITE  INTEGRALS 


267 


32.   Show  that  the  vokime  (hyperboloid  of  revolution)  generated  by 
revolving  about  the  x-axis  the  arc  of  the  hyperbola 


x^      y^  _  -J 


which   lies  in  the  first  quadrant,  and   is  terminated   by  the   ordinate 
X  =  c,  \^ 


Art.  10.  The  rectification  of  curves.  To  find  the  length 
of  an  arc  of  a  curve  is  equivalent  to  finding  a  straight  or 
right  line  of  equal  length,  into  which  the  arc  could  be 
straightened  out.  The  process  is  therefore  called  rectifying 
the  curve. 

Let  it  be  required  to  find  the  length  of  the  curve  y=/(a;), 
between  the  ordinate  x=  a  and  x  =  b. 

In  the  figure  (Fig.  54),  let  OQj^  =  a  and  OQ^^  =  b.     Then 
we  seek  the  length  of  the  arc  PiJ^^-     Divide  Q1Q2  up  intc 
71  equal  parts  each  of  length  Ax. 
Let  PB  =  A^,  then  the  chord 


F'F=\lAx^  +  Ag'' 


or 


prp 


-V 


l+lgj.i.. 


The  length  PH  or  Ay  of  course 
varies  at  different  points  along  the  Fig.  m. 

curve,  and  the  sum  of  the  hypote- 
nuses P'P  (corresponding  to  the  n  divisions  of  Q1Q2)  is  ^^^ 
approximation  to  the  length  of  the  arc.     This  approxima- 
tion is  the  closer,  the  larger  the  number  of  divisions  ;  i.e.  the 
smaller  Ax  is  taken.     The  actual  length  of  the  arc  itself  in 


268  CALCULUS  [Ch.  VIII. 

the  limit  which  this  sum  approaches  as  Ax  approaches  zero  ; 
or,  denoting  the  arc  by  s, 


lim 


-i:V'-(!fT-"j:v>Hi)'- 


Example.     To  Jind  the  circumference  of  a  circle. 

The  equation  of  the  circle  referred  to  the  center  as  origin  is 


\ 


x^  +  y'^ 

=  r\ 

Differentiating  with  respect  to  x, 

2x  +  2y'^  =  0, 

"t- 

y 

We  have  to  find  then  the  value  of 

CJl+^dx,    or  of 

fV.- 

x'^ 

.2  _  ^5 

-dx. 

We  find  the   length  of  one  quadrant   by.  taking  the   limits   of  the 
integral  0  and  r,  so  that  we  have  to  evaluate 


^'      ^^^        ^rTarcsin-^r     ^ 


J\ dx,   or   r\ 
0  \V2  _  a;2  Jo 


Vr2  -  x^        lo     .         r  2 

This  being  the  length  of  one  quadrant,  the  length  of  the  whole  circle  is 

2irr. 

EXERCISES    XXVII 

1.  Find  the  length  of  the  curve  (a  catenary) 

from  the  ordinate  x  =  0  to  the  ordinate  x  =  a.  Ans.  s  =  |(e«  —  e-«). 

2.  Find  the  length  of  the  curve  (a  semi-cubical  parabola) 

Z/2  —  x^ 

from  the  ordinate  :r  =  0  to  the  ordinate  x  =  a. 


10-11.]  DEFINITE  INTEGRALS  269 

3.    Find  the  length  of  the  curve  (a  cycloid) 

y  =  a  arc  cos ^ — I-  v  2  ax  —  x^ 


a 


between  the  same  ordinates  as  in  the  previous  exercises.  Ans.  2  aV2. 
4.   Find  the  length  of  the  curve  (a  hypo-cycloid) 

a;3  +  ?/3  =  a3 

between  the  same  ordinates  as  above.  Ans.  s  =  — • 

Art.  11.  Definite  and  indefinite  integrals.  The  connection 
between  definite  and  indefinite  integrals,  which  we  discussed 
above  (p.  253),  shows  that  the  absence  of  an  undetermined 
constant  of  integration  is  an  advantage  of  calculations  per- 
formed with  definite  integrals.  Indeed,  in  the  solution  of 
the  examples  which  we  have  treated  by  indefinite  integrals, 
definite  integrals  might  have  been  used  from  the  outset. 
Corresponding  to  the  indefinite  integral, 

(1)  ff<ix)dx  =  F(ix% 

we  have  the  definite  integral, 

Cyix-)dx  =  Fix^)-FCri}. 


X2 
Xi 

By  substituting  u  for  F(x)^  this  equation  may  be  written 

X2 
Xx 


(2)  u^-u^=  C  /(x)  dx, 


where  u^  and  u^  are  the  values  of  F(x)  corresponding  to  x^ 
and  x^.     Moreover,  the  differentiation  of  equation  (1)  gives 

(3)  !=/(.). 


270  CALCULUS  [Ch.  VIII. 

In  the  inverse  process,  we  pass  from  equation  (3)  to 
either  equation  (1)  or  equation  (2),  by  using  the  method 
of  indefinite  or  of  definite  integration. 

In  conclusion,  we  make  brief  illustrative  application  of 
definite  integrals  to  some  of  the  problems  which  we  have 
already  treated  by  use  of  indefinite  integrals. 

I.  We  found  the  equation  for  the  inversion  of  sugar 
(p.  182)  to  be 

dt  1 


or 


dx     K(a  —  x) 

If  the  values  t^  and  t^  correspond  to  the  values  x^  and  x^^ 
the  properties  of  definite  integrals  give  us  at  once 


.       ^        I  ^       dx  1 


K{a  -x)      K 


log 


a  —  x 


K     ^{a-x^ 

II.    In  considering  the  attraction  of  a  homogeneous  rod 
on  a  point  m  lying  in  its  direction,  we  found  the  following 

equation  (p.  219)  : 

dFjx^  ^     mM 
dx       (a  +  xy^ 

Now,  since  at  one  end  of  the  rod  x  =  0  and  at  the  other 
X  =  l^ 


_  r^  mMdx  _ 
Jo  (a  +  xf 


mM  ^n         1 

—  =  mM[ 


a  -\-  X  \a      a  +  I 


a  result  already  found  (p.  220)  according  to  the  methods 
of  indefinite  integration.     The  definite  integral. 


11.]  DEFINITE  INTEGRALS  271 

^_  r^  mMdx 
Jo    (a  +  xy^ 

gives  the  total  attraction  of  the  rod  as  the  limit  of  the  sum 
of  the  attractions  exerted  by  each  of  the  small  constituent 
parts  into  which  the  rod  is  arbitrarily  divided. 

III.  In  like  manner  it  appears  that  the  altitude  ^  above 
the  earth's  surface  (p.  223)  corresponding  to  the  atmos- 
pheric pressure  y  is  given  by  the  definite  integral, 


^=-x 


S   y-8    °^2,' 


IV.  Similarly  (p.  226),  the  time  elapsing  during  the  cool- 
ing of  a  body  from  the  temperature  6-^  to  the  temperature  6 
is  given  by  the  definite  integral, 

r'rnc     dO     ^rm^      6^-6^^ 

je,  K e-e^    K    ^  e-e^ 

Which  of  these  two  modes  of  calculation  is  to  be  chosen 
in  any  particular  problem  depends  upon  circumstances ; 
they  are  in  essence  but  slightly  different,  and  either,  cor- 
rectly applied,  leads  to  the  desired  result. 


19 


CHAPTER  IX 

HIGHER   DERIVATIVES   AND   FUNCTIONS   OF   SEVERAL 
VARIABLES 

Art.  1.  Definition  of  higher  derivatives.  The  derivative 
of  the  function 

(1)  y  =  sin  X 
has  the  value 

(2)  y'  =  ^  =  cos  X. 

ax 

This  derivative  is  likewise  a  function  of  a;,  and  its  deriva- 
tive is 

ax      ax  \axj 

The  expression  thus  obtained  is  called  the  second  deriva- 
tive  (or  the  second  differential  coefficient)  of   sin  x^  and  is 

denoted, by  y"  or  — ^.     We  have  therefore  the  equation 

The  second  derivative  is  also  a  function  of  x^  so  that  we 
can  form  its  derivative,  and  the  process  can  be  continued 
indefinitely.  That  which  has  just  been  said  concerning  the 
successive  derivatives  of  sin  x  may  at  once  be  extended  to 
all  the  functions  which  we  have  considered.     The  notation 

272 


1-2.]  HIGHER  DERIVATIVES  273 

is  analogous  to  the  preceding.     Considering  the  function  ^, 
or  /(:??),  the  expressions 

(5)  y',  y",  y"',  ....  or  /'(a?),  fix),  f"'(a^),  ..., 
or  also 

(6)  1^'  ^2'  —%'  "'  ^^'  ;^^^^^'  :r^^^^^'  ^/w,  -, 

denote  the  first,  second,  third,  etc.,  derivatives  of  our  func- 
tion ;  in  the  aggregate,  they  are  called  higher  derivatives. 

Just  as  —  has  been  used  as  a  symbol  for  the  operation  of 
differentiating,  with  respect  to  x^  that  which  stands  after  the 
symbol,  so  -—  is  a  symbol  for  the  operation  of  differentiat- 
ing twice  with  respect  to  x  that  which  stands  after  the 
symbol ;  and,  in  general,  — -^  denotes  the  operation  of  differ- 

(XiX 

entiating  with  respect  to  x^  n  times.     These  symbols  are  in 
no  sense  fractions. 
We  have,  for  example 

(7)  y,=g^^=^»(,),  etc. 


Akt.  2.  The  higher  derivatives  of  the  simplest  functions. 
I.  The  higher  derivatives  of  the  function  e^  are  the  simplest 
of  all.     Putting  y  =  e^,  we  have  (p.  144) 

y  =  ^^,   y"  =^  =  e^,   y  =  ^  =  g^,  etc. ' 

(XX  (XX 

All  the  derivatives  are  therefore  equal  to  each  other  and 
to  the  original  function  e^  itself.  Here  again  the  great 
simplicity  of  the  exponential  function  appears  clearly. 


274  CALCULUS  [Ch.  IX. 

II.    Considering  next 

y  =  cos  x^ 
we  have  (p.  124), 

/  'ft      dy^ 

y'=  —  smx^    y"  =  ~f- =  —  cos  x, 

dx 

yfff  =  _sZ_  =  sin  X,   y^^  =  -^ —  =  cos  x^  etc. 
dx  dx 

The  fourth  derivative  is  equal  to  the  original  function 
y ;  the  next  succeeding  derivatives  therefore  have  succes- 
sively the  same  value  as  y'  and  the  derivatives  following 
it.  The  same  law  holds  for  the  function  sin  x.  From 
y  =  sin  X  we  find  by  repeated  differentiation 

,  ,,      dy' 

y'  =  cos  x^   y    =  -f-  =  —  sm  x^ 

(X/X 


.fff 


dy'^  iv      dy'" 


y'"  =  -f—  =  —  COS  x^    y  =  -^ —  =  sin  x. 
dx  dx 

The  fifth  derivative  is  equal  to  cosic;  i.e.  equal  to  the 
first  derivative,  and  consequently  the  values  of  the  deriva- 
tives are  repeated  in  regular  sequence. 


III. 

Taking  up  next 

y 

=  log  a;. 

we  have  (p.  141) 

y' 

_l_^_i 

X 

and  (p. 

155) 

y" 

dx 

1  x-''  =  - 

1 

y,n 

dx 

.  2  ^-=^  = 

1  .2 

r 

_dy^^' 
dx 

=  -1-2. 

Z'X-^  = 

1  . 

2  • 
x^ 

3 

etc.,  etc. 


2.]  HIGHER  DERIVATIVES  275 

IV.    We  consider  finally 

2/  =  ^^ 

where  n  is  any  arbitrary  constant.     We  obtain  (p.  155) 
y'  =  nx"~^, 
y"  =  n(n  —  V)x^~'^^ 
y'^'  =71(71 -l^(n-2)x''-\ 
y'""  =  7i(n  -  l)(n  -  2)(n  -  S')x''-\ 
etc.,  etc. 

If,  in  particular,  n  is  a  positive  integer,  the  exponent  of  x 
in  the  (n  —  l)st  derivative  takes  the  value  7i  —(n  —  1),  or 
unity,  and  we  have 

y(n-i)=:n(n-l)(n-2)"-2'X, 

yW  =  ^(^  _  1)(^  _  2)  ...  2  .  1, 

and  since  y^^^  is  a  constant,  all  the  succeeding  derivatives 
are  zero.  If  n  is  not  a  positive  integer,  the  sequence  of  tlie 
derivatives  can  be  extended  as  far  as  we  please  without 
reaching  one  whose  value  is  zero. 

EXERCISES    XXVIll 

Find  the  second  and  the  third  derivative  of  each  of  the  following 
functions :  , 

1.    // =  e«*.  5.   y  =  sm^x.  q        _  1  —  a: 


2.  y  =  sin  ax.  1  +  x 

6.   w  =  xe*. 

3.  7/  =  e^  sm  X.                        ^                                     ^  x^ 

9.    2/  = 

4.  y  =  sin^  X.                     7.   y  =  x^  log  x.  1  —  x 

Find  the  nth  derivative  of  the  following : 

10.  y  =  e-'.  ^^    y=^—' 

11.  y  =  sin  ax.  1  -{-  x 

12.  y=(a  +  x)i  ^5     y  =  ^■ 

13.  y  =  log  x^.  Ve* 


276  CALCULUS  .       [Ch.  IX. 

Art.  3.   Geometric  meaning  of  the  second  derivative.     The 

second  derivative  has  an  important  geometric  signification. 
The  curve  (Fig.  bb)  is  said  to  be  concave  toward  the  a;-axis 
in  the  part  ABC,  and  convex  toward  the  a;-axis  in  the  part 
OBU.     Let 

(1)  ^=/(^) 

be  the  equation  of  the  curve,  and  let  us  turn  our  atten- 
tion to  the  values  which  tan  t  assumes  along  the  various 

parts  of  the  curve  (p.  123).  Be- 
ginning at  A,  the  angle  r,  and  like- 
wise tan  T,  continually  decreases  as 
the  •  point  describes  the  arc  AB. 
At  the  highest  point  B  of  the 
curve,   the    angle    t    is    zero,    and 

— ■ ^X      beyond  B,  r  is  but  little  less  than 

-p      ^^  TT,  and  again  decreases  continually 

until  the  moving  point  reaches  C; 
at  the  point  B,  tan  r  is  zero,  and  beyond  B  it  has  very  small 
and  continually  increasing  negative  values  along  BO;  i.e. 
tap  T  decreases  along  BC  also.  For  this  entire  arc  ABC  the 
derivative  of  tan  r  must  therefore  be  negative  (p.  125);  that 
is,  we  must  have 

C0\  ^  ^^^  '^  _  ^^'  _  ^V  ^  0 

dx  dx      dx'^ 

The  opposite  is  true  for  the  arc  CDE.  From  C  to  D, 
tan  T  has  negative  values,  which  decrease  numerically  to 
zero,  and  from  D  to  E^  tan  t  assumes  continually  increasing 
positive  values.     Therefore,  for  this  part  of  the  curve, 

C^\  <^.tan  T  _  d?y      r. 

dx  doiP' 


3.]  HIGHER  DERIVATIVES  277 

We  see  thus  that  the  sign  of  the  second  derivative  indi- 
cates whether  the  curve  is  concave  or  convex  toward  the 
a;-axis. 

The  two  arcs  of  the  curve  are  separated  by  the  point  (7,  in 
which  the  second  derivative  becomes 

(1)  ^'0: 

and  this  is  called  a  point  of  inflexion. 

The  convexity  or  concavity  of  a  curve  naturally  depends 
upon  the  side  from  which  the  curve  is  viewed.  As  our 
figure  was  constructed  above,  the  curve  lay  entirely  on  the 
positive  side  of  the  2:-axis.  If  the  curve,  on  the  other  hand, 
lies  entirely  on  the  negative  side  of  the  rc-axis  (the  a:-axis 
may  be  imagined  to  be  moved  parallel  to  itself  until  this 
is  the  case),  the  part  of  the  curve  which  was  formerly  con- 
vex toward  the  a:-axis  is  now  concave,  and  vice  versa.  To 
obtain  a  complete  criterion  for  the  direction  of  curvature 
of  the  curve,  we  notice  the  following :  In  Fig.  55  the  arc 
ABC  is  concave  toward  the  a;-axis;  for  it,  i/  is  positive  and 

— ^  is  negative ;  in  the  altered  position  of  the  a;-axis,  CBU 

is  concave  toward  the  rr-axis ;  for  it,  y  is  negative  and  the 
second  derivative  positive ;  in  both  cases,  therefore. 

For  convex  arcs  the  reverse  is  true.  Along  the  arc  CDE 
in  Fig.  3b.,  both  y  and  the  second  derivative  are  positive, 
while  in  the  altered  figure  both  these  quantities  are  nega- 
tive for  the  arc  ABC^  and  therefore  in  both  cases 


278  CALCVLVS  [On.  IX. 

We  have  thus  obtained  a  complete  criterion  to  determine 
whether  any  arc  of  a  curve  is  convex  or  concave  toward  the 
X-axis. 

A  simple  example  is  offered  by  the  sine  curve  (Fig.  39, 
p.  123),  whose  equation  is 

y  =  sin  X. 

The  second  derivative  is  alternately  negative  or  positive, 
while  the  corresponding  y  is  alternately  positive  or  nega- 
tive ;  the  curve  is,  therefore,  always  concave  toward  the 
rr-axis.  The  intersections  of  the  curve  with  the  a;-axis  are 
all  points  of  inflexion. 

If  we  select  the  point  of  view  below  the  curve  (the  direc- 
tion in  which  a  point  with  constant  abscissa  moves  when  its 
ordinate  is  diminished,  being  regarded  as  downward),  we 

may  also   say  :    If  — ^  is  positive  for  any  value  of  a;,   the 

d^u 
curve  is  convex  downward  for  that  abscissa,  while  if  —4 

dx^ 
is  negative,  the  curve  is  concave  downward.* 

Art.  4.  Physical  interpretation  of  the  second  derivative. 
Like  the  first  derivative,  the  higher  derivatives  also  are  of 
great  importance  in  the  applications  to  physical  science. 
For  our  purposes  it  will  suffice  to  make  the  meaning  of 
the  second  derivative  clear  by  several  examples. 

Let  any  rectilinear  motion  (^e.g.  that  of  a  freely  falling 
body)  be  given  by  the  equation 

(1)  i=fcf). 

Suppose  that  in  t,  t^^  t^^  ^^  "' 


*  An  exception  may  arise  when  -^  =  0  or  co  for  the  value  of  x  in  ques- 

dx 
tion.     This  case  will  be  taken  up  in  another  connection  (p.  362). 


a-4.]  IIIQHMR  DERIVATIVES  279 

units  of  time  Qe.g.  seconds)  from  the  beginning  of  the 
movement,  the  moving  point  has  passed  over 

units  of  length  (^e.g.  meters),  respectively,  and  let  its  velocity 
at  these  times  be,  respectively, 

V,   Vp    V^,   V^,    '". 

The  velocity  will,  in  general,  change  each  instant.  To  form 
an  idea  of  the  nature  of  this  change,  we  introduce  the  well- 
known  concept  of  acceleration.  Considering  first  uniformly/ 
accelerated  motion^  i.e.  motion  in  which  the  velocity  receives 
equal  increments  in  equal  intervals  of  time,  the  acceleration 
is  the  increase  in  velocity  which  takes  place  in  a  second.  If 
in  T  seconds  the  increase  in  velocity  has  been  t)  meters,  and 
if  a  denote  the  acceleration,  we  have 

(2)  •  «  =  J 

We  now  apply  this  idea  to  the  motion  represented  by 
equation  (1),  which  is  not  uniformly,  though  continuously, 
accelerated,  i.e.  the  increase  in  velocity  is  not  always  the 
same  in  equal  intervals  of  time  at  different  periods  in  the 
motion,  but  there  is  no  instantaneous  increase  due  to 
the  impetus  of  a  new  force  suddenly  applied.  As  there 
is  no  perceptible  instantaneous  increase  at  any  point  in  the 
motion,  the  amounts  of  increase  in  velocity  in  consecutive 
time  intervals  approach  more  and  more  nearly  to  equality 
as  the  intervals  of  time  are  taken  smaller  and  smaller,  and 
the  limit  which  the  ratio  of  the  time  interval  to  the  incre- 
ment of  velocity  approaches  as  the  former  is  diminished, 
is  b}^  (2)  the  acceleration.  If  Az;  denote  the  increase  in 
velocity  in  the  time  A^,  then  the  limit  of  the  ratio  —  as  Ai 


280  CALCULUS  [Ch.  IX. 

approaches  zero  is  the   acceleration.     But  this  limit  is  the 
derivative  of  v  with  respect  to  t^  and  hence 

®  "I- 

We  have  seen  in  (p.  109),  that  v  is  the  derivative  of 
I  with  respect  to  the  time,  ^^g., 

(4)  v=f, 

at 

and  hence 

(5)  «  =  ^  =  ^. 
^  ^  dt      dt^ 

For  instance,  we  have  for  the  motion  of  free  fall  (p.  167), 

v  =  gt, 

and  hence  the  acceleration  is 

dH  __dv  _ 
dr^~di~^' 

This  result  shows  that  the  acceleration  is  a  measure  of  the 
force  which  causes  the  motion ;  and  in  fact,  we  may  define 
forces  by  means  of  the  accelerations  which  they  impart  in 
the  unit  of  time  to  a  body  of  unit  mass  on  which  they  act. 
If,  therefore,  the  law  of  any  motion  is  known,  we  can  deter- 
mine what  the  forces  are  which  cause  the  motion. 

Art.  5.  Oscillatory  motion.  Let,  for  example,  a  point  P 
of  unit  mass  move  upon  a  straight  line  (Fig.  6Q^  so  that  its 

distance  x  from  a  fixed  point  0  is 


M 

0          p 

Fig.  56. 

N 

given 

(1) 

by 

the  equation 
x  =  A  sin  t. 

For 

the  velocity 

of  the  point 

we 

have 

(2) 

V  - 

dx 
dt 

A  cos  t, 

4-5.]  HIGHER   DERIVATIVES  281 

and  from  this  by  forming  the  second  derivative  we  have  for 
the  acceleration 

Substituting  for  A  sin  t  its  value  x  from  equation  (1),  we 
have 

This  equation  states  that  the  acceleration,  and  therefore 
the  force  acting  on  the  pointy  is  equal  to  its  distance  from  the 
fixed  center.  The  negative  sign  denotes  that  the  tendency 
of  the  force  is  to  diminish  the  distance  of  the  movable  point 
from  0 ;  that  is,  the  force  is  attractive. 

Under  this  law  are  included  most  of  the  motions  in  which 
bodies  oscillate  about  a  center  of  equilibrium.  Such  are, 
according  to  Huygens,  the  motion  of  the  particles  of  ether 
in  the  propagation  of  light,  the  motion  of  the  particles  of 
air  in  the  propagation  of  sound,  the  vertical  motion  of  the 
particles  of  water  in  the  propagation  of  water-waves,  the 
movement  of  the  particles  of  a  vibrating  string ;  in  short,  all 
those  movements  of  small  particles  which  take  place  in  the 
motion  of  either  stationary  or  progressive  waves. 

The  nature  of  this  motion  is  like  that  of  the  pendulum 
about  its  position  of  rest.     If  for  t  we  put,  respectively, 

"'    2'   '''   T'   ^'''   ~Y'    ■■■' 

then  we  have  for  x^  v,  a,  at  the  times  just  mentioned,  the 

values 

X  =  0,        A,         0,    -A,      0,        A,     •••; 

V  =A,         0,    -A,         0,     A,         0,     ...; 

a=  0,    -A,         0,       A.      0,    -yl,     ..». 


282  CALCULUS  [Ch.  IX. 

These  results  enable  us  to  sketch  the  progress  of  the 
motion.  At  the  beginning  of  the  motion,  i.e.  at  the  time 
^  =  0,  the  point  is  at  0  and  has  the  velocity  A^  while  the 
active  force  is  zero.      After  —  seconds  the  point  is  at  iV, 

its  velocity  has  in  the  meantime  been  reduced  to  zero, 
while  the  attractive  force  has  reached  its  greatest  value,  A. 
The  point  now  reverses  the  direction  of  its  motion,  and  in 
consequence  of  the  attractive  force  its  velocity  increases 
constantly.  In  ir  seconds  the  point  is  again  at  0,  and 
passes    through   0  with   the    highest  velocity   that    it    can 

o 

attain.     In  -—  seconds  the  point  is  at  il[f,  and  has  attained 

its  greatest  distance  from  0  toward  the  left ;  its  velocity 
is  again  zero,  while  the  force  has  again  reached  its  maxi- 
mum value.  In  2  TT  seconds  the  point  is  again  at  0,  passes 
through  0  with  the  velocity  J.,  and  the  attractive  force 
is  zero.  The  latter  now  increases  again,  while  the  velocity 
constantly  diminishes  until  the  point  reaches  N  again,  and 
thus  it  continually  and  regularly  swings  back  and  forth  in 
a  straight  line. 

Art.  6.  The  velocity  acquired  by  a  body  falling  toward 
the  earth  from  a  great  distance.  We  suppose  that  the  body 
falls  from  rest,  under  the  influence  of  the  earth's  attraction 
alone.  As  the  body  falls  from  a  great  distance,  we  may  not 
suppose,  as  heretofore,  that  the  acceleration  is  constant,  but 
must  take  into  account  the  law  that  the  acceleration  varies 
inversely  as  the  square  of  the  distance  between  the  attract- 
ing bodies.  Let  a;  and  x^  denote  the  distance  of  the  body 
from  the  earth's  center  at  the  time  t  and  at  the  time  the  fall 
began,  respectively,  and  let  g  be  the  constant  of  gravity  at 
the  earth's  surface,  and  r  the  radius  of  the  earth.  Then,  as 
we  have  already  seen,  (p.  109),  the  velocity  at  this  time  is 


5-6.]  HIGHER  DERIVATIVES  283 

'Q  —  tJC) Q/ou 

dt       ~~'dt' 
df 


,  ^  ^  d{xQ  —  x)^      dx 

and  the  acceleration  is 

(2)  a^ 


dx 
Multiplying  both    members   of  (2)   by  —  2—-,   and    mte- 

grating  with  respect  to  t,  we  have 

(3)  f-2apt^{fj^^. 

The  velocity  and  the  acceleration  are  only  apparently 
negative  since  the  variable  x  is  decreasing,  and  hence  the 
first  derivative  is  itself  negative,  while  by  its  definition,  V  is 
increasing.  The  acceleration  at  the  earth's  surface  is  g,  and 
by  Newton's  law  of  gravitation. 

From  (3)  and  (4), 

(6)  But       f-tif^it.f-^d^.hs^+a. 

^         x^    dt         ^^         x^  X 

or, 

(7)  ^;2  =  2^+(7. 

x 

When  the  fall  began,  v  =  0,  so  that 

whence, 

(8)  ^  =  2^'-<i-D- 


284  CALCULUS  [Ch.  IX. 

This  formula  gives  the  velocity  at  the   distance  x  from 
the  earth's  center.     At  the  earth's  surface, 

1       1 


(9)  ?;2  =  2  gr 


r      x^ 


If  the  distance  x^  is  increased,  v^  approaches  the  limit  2^r. 
No  matter  from  what  distance  a  body  falls  to  the  earth 
(under  the  influence  of  the  earth's  attraction  alone),  its 
velocity  on  reaching  the  earth  will  always  be  less  than 
V2^r,  which  may  readily  be  found  to  be  not  quite  7  miles 
per  second.  Disregarding  the  resistance  of  the  earth's 
atmosphere,  we  see  that  if  a  body  were  projected  vertically 
upward  with  the  initial  velocity,  ^V2^r  (e.g.  seven  miles 
per  second),  it  would  never  fall  back  to  the  earth. 

In  a  similar  manner,  the  limit  of  the  velocity  with  which  a  falling- 
body  would  reach  other  bodies  can  be  determined.  For  the  sun,  the 
velocity  is  about  383  miles  per  second.  A  body  passing  the  earth's  orbit, 
under  the  influence  of  the  sun  only,  would  be  moving  at  about  26  miles 
per  second.  This  is  nearly  the  average  velocity  with  which  meteors 
enter  our  atmosphere. 

The  velocity  needed  to  project  a  body  beyond  the  range  of  the  moon's 
attraction  is  small ;  it  has  been  supposed  that  the  moon  lost  its  atmos- 
phere for  this  reason.  For  the  sun,  on  the  other  hand,  the  velocity  is 
very  great,  and  the  sun  retains  an  enormous  atmosphere. 

The  formula  (9)  can  also  be  used  in  fall  through  a  short  distance  if 
we  wish  to  take  into  account  the  variation  of  the  earth's  attraction,  since 
we  introduced  no  hypothesis  as  to  the  magnitude  of  x^  until  after  (9) 
was  established.  For  a  small  distance  the  difference  between  the  velocity 
computed  by  (9)  and  that  found  on  the  assumption  that  the  earth's 
attraction  remained  constant  would,  of  course,  be  very  slight. 

Art.  7.  Partial  derivatives.  If  a  gas  is  subjected  to 
a  variable  pressure  and  temperature,  its  volume  v  is  depend- 
ent upon  the  pressure  'p  as  well  as  the  temperature  Q.  We 
say,  therefore,  that  v  is  a  function  of  'p  and  of  Q.  Similarly, 
the  area  of  an  ellipse  is  a  function  of  its  semi-axes,  a  and 
5 ;  the  volume  of  a  parallelopiped  is  a  function  of  its  three 


6-7.]  FUNCTIONS   OF  SEVERAL    VARIABLES  285 

edges,  etc.  These  are  examples  of  functions  of  two  or  more 
variables.  In  correspondence  with  our  previous  nomen- 
clature (p.  112),  we  call  the  volume  v^  regarded  as  a  func- 
tion of  p  and  ^,  the  dependent  variable,  while  we  call  p  and 
6  the  two  independent  variables;  similarly  in  the  other 
instances  mentioned. 

Denoting  the  variables  by  x^  y,  2,  •••,  the  following  symbols 
have  been  introduced  for  functions  of  two  or  more  variables  : 

f(x,  ?/),  F(x,  y,  z),  (pCx,  «/), ...,  etc. 

We  now  extend  the  rules  and  theorems  of  the  differential 
calculus  to  these  functions. 

We  begin  with  the  following  illustration  :  Let  S  be  the 
area  of  the  rectangle  OABC  (Fig.  57),  whose  sides  have  the 
lengths  X  and  «/,  so  that  o'  c  c' 

(1)  S  =  xy.  B 

Regarding  the  lengths  of  these 
sides  as  variable,  the  area  S  of  the 
rectangle  will  vary  with  them.     We 


D 


first  vary  the  rectangle  by  altering      0  A  A 

only  the  length  x  of  the  side  0^, 

and  leaving  the  side  OB  =  y  fixed.  Regarded  so,  the  area 
aS'  is  a  function  of  x  alone,  and  we  can  apply  to  it  all  the 
rules  of  the  Differential  Calculus  previously  explained. 
We  assign,  therefore,  to  the  side  OA  —  x  an  increment  AA! ., 
which  we  represent  by  Aa;,  and  determine  the  carresponding 
derivative  of  S.     To  show  that  we  now  regard  aS'  as  func- 

tion  of  X  alone,  we  denote  this  derivative  by  — ^^-^,  and  by 

dx 
differentiating  equation  (1)  we  have 

(2)  ^^M  =  ^^ 

dx 


286  CALCULUS  [Ch.  IX. 

The  rectangle  OABG  can  also  vary  so  that  the  length  of 
the  side  OB  is  changed,  while  on  remains  constant ;  the  area  8 
is  then  a  function  of  y  alone.  We  assign  now  to  y  an  incre- 
ment Ay,  and  in  this  case  denote  the  derivative  by  —7^ ;  if 

under  these  assumptions  we  differentiate  equation  (1)  with 
respect  to  y,  we  find 

(3)  ^^  =  ^. 

dy 

The  notation, 

(4)  .  —  and  — , 
^  ^  dx  dy 

has  been  introduced  for  the  two  derivatives  above,  in  which 
the  round  d  is  used  to  indicate  that  we  regard  S  in  one  case 
as  a  function  of  the  one  variable  x  only,  and  in  the  other  case 
as  a  function  of  the  variable  y  only.     We  have,  therefore, 

dx  dy 

What  is  said  above  for  the  derivative  of  the  area  of  a 
variable  rectangle  may  be  extended  to  all  functions  of  two 
independent  variables.     Let 

be  such  a  function.  We  may  first  suppose  that  x  varies  but 
y  remains  constant,  or,  in  other  words,  we  may  regard  i6  as  a 
function  of  x  alone ;  if  upon  this  hypothesis  we  differentiate 
u  with  respect  to  x,  we  obtain  the  partial  derivative  of  u, 
with  respect  to  x^  the  notations  for  which  are 

^  and  ^f(^iJLl, 
dx  dx 

the  round  3's  serving  to  remind  us  constantly  that  we  are 
dealing  with  partial  derivatives. 


7.]  FUNCTIONS   OF  SEVERAL    VARIABLES  287 

Similarly,  if  we  suppose  u  to  be  a  function  of  y  alone,  we 
obtain,  by  differentiating  with  respect  to  ^, 

^,  or  ^fi^il}^ 
dy'  by 

the  partial  derivative  of  u  with  respect  to  ^. 

EXAMPLES 
1.    Let   w  =  ax2  +  c7/2,  ^    wz^sin^. 


6w 


y 


then  —  =  2  aa;,  . 

^x  To    dmerentiate    this    we    put 

Qu      ^  -  =  z,  then 

and  -—  =  2cy.  y 

^y  u  =  sin  z, 

2.  u  =  ^^-2,2.  and  3«^rf«5£ 

dx      dz  dx 

^=2x;    —  =  -2y.  =  cos  z  - - 

ox  ay  y 

1       x 
=  -cos— 

3.  M  =  sin  X  cos  ?/•  *  ^  y       y 

Similarly, 

-^  =  cos  :r  cos  y;    ~=-  sin  x  sin y.  du      du  dz 

dx  dy  - 


4.    M  =  log(2:2  +  y^).  =cosz  ■       ^ 


dy     dz  dy 

2 


du         2x        du 


r 


cos- 


dx     x'^  +  y'^     dy     x^  +  ?/2  y^       y 

The  above  definitions  and  notations  may  be  immediately 
extended  to  functions  of  three  or  more  independent  variables. 

If  u  =f(x,  y,  z) 

be  a  function  of  the  three  independent  variables  x^  y^  2,  then 
we  have  the  three  partial  derivatives, 

^,  or  ^fC^ilill,  ^,  or  ^f(^ilil), 

dx  dx  dy  dy 

and  ^,  or  ^S^^^, 

dz  dz 

■  m 


288  CALCULUS  [Ch.  IX. 

in  which,  in  each  case,  the  definition  requires  that  the  vari- 
able with  respect  to  which  the  differentiation  is  made,  be 
regarded  as  the  only  one  to  vary. 

Art.  8.  Higher  partial  derivatives.  Functions  of  two 
or  more  variables  may  be  differentiated  repeatedly  with 
respect  to  any  or  all  of  the  variables.  The  notation  used 
is  the  following : 

— -  denotes  the  result  of  differentiating  u  twice  with 
respect  to  a?,  y  being  treated  as  a  constant ; 

— ^,  —  denote  the  result  of  differentiating  similarly  three 

or  n  times ; 

— —  denotes   the   result   of   differentiating  u  first  with 
dy  dx 

regard  to  x^  and  then  differentiating  that  result  with  regard 
to  y; 

—  denotes   the   result    of   differentiating   twice   with 

dx  by^ 
regard   to   y,  then  with   regard   to  x.      Similar   notations, 

understood  without  difficulty,  are  used  for  still  more  differ- 
entiations. % 
Let  us  take,  for  example, 
(1)                                       It  =  sin  a:  cos^  ^. 

1  hen    —  =  cos  x  cos^  y^     —  =  —  zsinx  cos  y  sm  y. 
dx  dy 

Differentiating  each  of  these  with  respect  to  x,  and  also 

with  respect  to  y^  we  obtain 

— -  =  —  sm  a;  cos^  yl  =  —  2  cos  x  cos  y  sin  y, 

dx^  ^  dxdy  '^        ^ 

^  =  —  2  cos  a;  cos  1/ sin  ^,      — ^  =  —  2  sin  a;  [cos^  ^  —  sin^ «/] . 


dy  dx  dy 


7-8.]  FUNCTIONS  OF  SEVERAL    VARIABLES  289 

We  notice  in  the  above  illustration  that  the  values  of  

and are  equal.     This  holds  true  as  a  sfeneral  theorem.* 

dxdy  ^  ^ 

It  may  be  made  evident  as  follows  : 

Consider  u  =f(x^  y) ;  we  wish  to  prove  that 

(2)  J^  =  _^. 

bybx      dxdy 

By  definition, 
/ox  ^^_     lim     f(x  +  Ax,y)~f(x,y^ 

/^\  d^u   _  _d_^du]  _     lim     j     lim 

^  ^  dydx~  dy\dxi~^y  =  ^{^'^  =  ^ 


fix  4-  Aa;,  ^  +  A^)  -f(x,  y  +  A^/)  -f(x  +  Aa;,  ^y)  -/(a:,  ,v)  j 

Aa;Ay  ) 

Similarly, 
,rx  du_     lim     .f(:r,  ,y  +  Ay)  -/(a;,  y) 

rl    /A^  ^^^    —   d  {du\  _     lim      (     lim 


./'(a:  +  A.T,  y  +  Ay)  — /(a:^  +  Ax,  y)  —  f(x.  y  +  Ay)  —f(x,  y) 

Ax  Ay 

A  slight  rearrangement  will  show  that  the  fractions  of 
which  the  limits  are  taken  are  the  same  in  both  cases,  the 
only  difference  in  the  two  expressions  being  that  Ax  and  Ay 
approach  zero  in  different  orders.  It 'seems  plausible  f  to 
think,  however,  that  the  final  results  are  the  same,  no  matter 

*  Of  course,  with  the  restriction  (which  we  always  tacitly  make)  that  the 
function  and  all  the  derivatives  concerned  are  continuous. 

t  The  rigorous  proof  of  this  theorem  is  too  difficult  to  find  a  place  here. 


290  CALCULUS  [Ch.  IX. 

in  what  order  Ax  and  Ay  are  made  zero.  For  in  the  end 
there  are  left,  as  the  limit  sought,  those  terms,  and  those 
only,  which  vanish  with  neither  Ax  nor  Ay.  The  same  limit 
is  therefore  found  in  each  case,  or 


doc  dy     By  doc 
As  an  immediate  consequence  of  this  result,  it  is  apparent 
that  the  order  of  any  number  of  differentiations  is  immate- 
rial.    Thus,  there  are  only  four  distinct  third  partial  deriva- 
tives of  u  above,  viz. : 

da^     dy  dx^     dy^dx     dy^ 

All  others  are  reducible  to  one  or  other  of  these  by  an 
interchange  of  order  of  differentiation  ;  thus, . 

d^u     _    d^u 
dx  dy  dx      dy  dx^ 

EXERCISES   XXIX 

Find  —  and  —  for  each  of  the  following  functions  : 
dx  dy 

1.  u  —  x^  -\-  y"^  —  a^.  4.    u  :=  x^  -{■  ax'^y  -\-  hxy'^  +  y^. 

2.  u  =  X  cos  y.  5.   u  =  x^  —  3  xy^  +  6  y*. 

3.  u  =  ye""  +  xe^. 

Hint.  —  In  the  following  four  exercises  use  the  method  of  p.  152. 

6.  M  =  Vx'^  +  y'\  8.    u  =  sin  (x^  —  y^) . 

«  

7.  u  =  log{x  -  ?/2).  9.    u  =  e^\ 

Verify  that  =  _?Uf_  for  each  of  the  following  functions  : 

dx  dy      dy  dx 

10.  u  =  xBmy.  ^^    u=cot^^Jl- 

X 

11.  u  =  xe^  log  y.  T  e  o   •        ,        •   2 

°^  15.   M  =  x-^  sm  2/ +  ?/ sm^x. 

12.  u  =  tan  x^y.  16.    u  =  y^. 

13.  M  =  5  0:6  + 13  ^3^4  + 12  x'^y  -  32  yK   17.   u  =  sin  v  xp. 


8-9.]  FUNCTIONS  OF  SEVERAL    VARIABLES  291 

Form  ^  „  ^    and  — -  for  each  of  the  following  functions : 
dy^  ox  dx^ 

18.  u  =  x(x^  +  y^).  22.   u  =  x". 

19.  u  =  ax^-j-  hxy  +  cy\  23.    u  =  log-- 
__               .  w 

20.  M  =  sin  x  cos  y. 

24.    M  = 


21.    M  =  ^-2^'.  '  a-{-a:2_^^2 

Art.  9.  Differentiation  of  a  function  of  two  or  more  func- 
tions of  a  single  independent  variable.  We  have  already 
(p.  152)  considered  the  differentiation  of  a  function  of  one 
function  of  a  single  variable,  and  have  found  the  following 
result : 

If  y=K^')  and  z  =  <^(x),  or  y=f\(i>(x)\, 

(1)  then  dy^d^^dz^^ 

dx      dz     dx 

We  shall  next  consider  the  case  of  a  function  of  two 
functions  of  a  single  variable,  or  of  two  variables  dependent 
upon  a  single  independent  variable. 

Accordingly,  let 

^  =f(:y^  2;), 

where  y  =  <l>{x)  and  z  =  "^(x). 

The  result  which  we  shall  prove  is 

^ON  du  _du  dy      du  <h 

dx      dy  dx      dz  dx 

In  words  :  The  derivative  of  the  given  function  with  respect 
to  the  independent  variable  is  equal  to  the  partial  derivative 
of  the  given  function  with  respect  to  one  of  the  dependent  vari- 
ables times  the  derivative  of  that  dependent  variable  ivith  respect 
to  the  independent  variable^  plus  a  similar  product  in  which 
the  other  dependent  variable  is  used. 


292  CALCULUS  [Ch.  IX. 

Letting    u  +  Att    be    the    value   which    u    assumes    when 

we  give  x  the  value  x  +  Aa;,  then,  by  the   definition  of  a 

derivative, 

^ON  du  __     lira      U  4-  Alt  —  2^ 

To  find  the  value  of  this  limit  it  is  necessary  to  determine 
u  +  ^u  more  closely. 

We  have         u=f{y,  z)=f\(i>(x),  'f(x)\. 

Replacing  a:  by  a;  +  Aa;, 

let  y  =  (f>(x)    become   y  +  A^  or  (^(x  -\-  Ax} 

and  s  =  yjr^x^  become    z  +  Afe  or  1/0(3;  +  Aa;), 

(4)  then 

u  +  Au=f\ (^(x  +  Ax),  yjr^x  +  Aa:) (  =f(y  -{■  Ay,  z  +  A2). 

Substituting  this  in  (3), 

,rx  (^^_    lim    f(y  +  A,z/,  g  +  Az)  -f(y,  z) 

^^^  ^.."^^  =  0  Ax 

Adding  and  subtracting  /(«/,  z  +  A2)  in  the  numerator, 


(^)    :r::-Ax  =  o 


^  Mm 

dx 

f(y  +  A,y,  2;  +  Az)  -f(y,  z  +  Az)  -j-f^y,  z  +  Ag)  -/(.y,  g) 

Aa; 


^    hm    f(y  +  A,y,  g  +  Az)  -fjy,  z  +Az) 


+ 


Urn    f(y,z-\-Az)-f(y,  z)_ 


Ax  =  0 


Ax 


Multiplying  numerator  and  denominator  of  these  fractions 
respectively  by  Ay  and  Ag, 


&.]  FlfJ^CTWNS  OF  SEVt:ttAL    VAUiABLES  29B 

n\      ^=    li"^    /(^/  +  A^,  g  +  Az)  -/(y,  g  +  Ag)     Ay 
^  "^       (^:r      ^^  ^  0  A?/  *  A2; 

-L     lini     /(y,  g  +  Ag)  -/(y,  g)      A^ 

■^  Ax  =  0  ^^  •  ^^• 

To  determine  the   value  of   these  limits  we  notice  that 
when  Ax  approaches  zero,  A?/  and  Az  approach  zero  likewise. 
We  may  write  then 

^oN      du_     lim    /(?/  +  A?/,  z  +  Az^  —  f(y,  z+Az)     lim    A^ 
^^     (^^      ^^  =  0  Ay  ^^  =  ^Ax 

,      lim    /(y.  g  +  Aa;")  —  /(y,  g)     lim     Az 
+  A.-0  ^^  Ax-0^^- 

The  various  factors  here  are  all  in  the  defining  form  for 
derivatives  except  the  first,  viz.  : 

rQ\  li™    /(y  +  Ay,  z  +  Az')-f{y,  z  +  Ag) 

This  would  be  in  the  defining  form  for  a  derivative  if  Az 

were  constant ;  namely,  it  would  be  — f  (y^  z  -\-  Az).      But 

dy 

under  the  conditions  of  our  problem,  Az  approaches  zero  at 

the  same  time  that  Ay  does  so,  so  that  ^  J^  0  (^  +  ^^)  ^^  ^' 
and  hence 

.-l^.       lim    /(y  +  Ay,  g  +  Ag) -/(y,  z  +  Az)  _  d 

We  use  the  round  5's  to  indicate  that  /(y,  z)  is  differ- 
entiated with  respect  to  y  alone. 

Substituting  this  in  (8),  and  replacing  the  other  factors 
by  their  limits,  we  obtain,  finally, 

due     dy  doc     dz     doc 


294  OALOuLm  [Ch.  IX. 

In  quite  the  same  way,  we  can  establish  similar  formulse 
for  functions  of  more  than  two  functions  of  an  independent 
variable.  Thus,  if  u  =/(^,  2,  10}^  ?/,  55,  and  w^  being  func- 
tions of  a:,  the  method  used  above  will  show  that 

^-j  Qx  du  _dudy      du  dz^      du  dw 

dx      dy  dx      dz  dx      div  dx 


EXAMPLES 

1. 

ler 

Let 
e 

Z^         ?/2 

z  =  a^^cosx, 
y  =  a%2  sin  X ; 

3n 

du 
dy~ 

fe2' 

du      2z, 
dz       a2' 

-M.  —  aHfi  cos  x\    ~  —  —  0^62  sin  x ; 
dx  dx 

—  =  — ^  •  aPlP-  cos  a:  +  ^^  (  —  a^fe^  gji^  x) 

dx  fc2  q2 

=  2  a^?/  cos  X  —  2  522  sin  x 
=  2a262sinxcosx(a2_&2). 


2.   Let 

u  =  (mf  +  fe2;2)» 

where 

?/  =  e"^;    2;  =  6-*; 

then 

^  =  n(ay^  +  bz^r-^-2ay; 
dy 

^^n{ay^  +  hz^Y-^'2hz; 

dz 

Hence      ^  =  2  anye^ay^  +  &22)n-i  _  2  hnze-Hay'^  +  &22)n-i 

The  same  result  would,  of  course,  be  reached  if  we  first  substitute  the 
values  of  y  and  z  in  u,  and  then  differentiate. 


9-10.]  FUNCTIONS  OF  SEVEMAL    VARIABLES  295 

3.  u  =  x^. 

This  function  involves  only  the  one  variable  x^  and  we  have  found 
its  derivative  before  by  passing  to  logarithms,  inasmuch  as  none  of  our 
fundamental  rules  for  differentiation  were  directly  applicable.  We  can 
also  differentiate  it  by  our  present  method,  regarding  it  as  given  thus : 

u  =  ?/^ 
where  y  =  ^^ 

z  =  x\ 

dy  dz 

djl  _  A.         dz  _  . 
dx  dx 

Hence  —  =  zy'-^  +  y"  logy,  or  substituting  the  values  of  y  and  2;, 
dx 

—  =  x"^  +  x""  log  X  =  x*(l  +  log  x). 

EXERCISES  XXX 

Find  —  for  the  following  functions  : 
dx  ^ 


1.    u  =  z^-\-  if. 

3.    u  =  z^f.     y  =  log  x;  z  =  x^. 

z  =  sin  x;  y  =  eosx. 

4.    It  =  z'^  +  yz  +  y^. 

2.    u  =  z\ogy. 

y  =  V^;  z  =  e'. 

5.   u=l'      y  =  x-',  z  =  e-. 

6.   Prove  equation  (12),  p.  294,  as  suggested  in  the  text. 

Art.  10.   Differentiation   of   implicit   functions.     It  often 
happens   that  we   have   an   independent  variable   x   and    a 
variable  «/  dependent  on  it,  for  which  the  relation  is  not 
expressed  in  the  form 
(1)  2/  =  ^(*),* 

*  The  symbol,  =,  means  "is  identical  with."  Identities  are  often  written 
with  the  usual  sign  of  equality,  as  we  have  done  hitherto  ;  but  the  use  of  the 
symbol,  =,  permits  us  to  emphasize  by  the  notation  (whenever,  as  in  the 
present  instance,  we  may  wish  to  do  so),  the  fact  that  we  are  dealing  with 
identities  and  not  with  equations  of  condition. 


296  CALCULUS  [Ch.  IX. 

but  in  the  form 

(2)  4>(^,i/)=o. 

In  this  case  ^  is  said  to  be  an  implicit  function  of  x,  and 
vice  versa^  while  in  the  relation  (1),  i/  is  said  to  be  an 
explicit  function  of  x. 

We  could  find  -^  by  first  solving  the  identity  (2)  for  i/  in 

Cl/X 

terms  of  x^  obtaining  y  as  an  explicit  function  of  x^  and  then 
differentiating  the  result.  In  many  cases,  however,  the 
identity  (2)  might  be  too  complicated  to  admit  of  solution ; 
in  many  other  cases  the  solution  may  be  possible,  and  yet 

not  convenient  to  effect  or  simple  in  form.     In  all  cases,  -— 

clx 
may  be  found  as  follows  : 

Let  w  =  <^(a;,  ?/),  in  which  we  regard  both  x  and  y  as  func- 
tions of  X ;  then  by  our  previous  results  we  have 

du  _  dudx      du  dy 
dx      dx  dx      dy  dx 

But  as  (2)  is  an  identity,  i^  =  0,  we  ma}^  equate  the  deriva- 
tives of  its  members,  and  have 

^ON  ^=0  du      dudy  _  r. 

dx  dx      dy  dx 

du 

Hence,   (4)  ||.-|. 

dy 
Introducing  the  value  of  u, 

d(t>(x.  y) 

(5)  ^^ ^^— . 

dx  d<l>{x,  y} 

dy 


10.]  FUNCTIONS  OF  SEVERAL    VARIABLES  297 

For  example : 

I.    Let  the  relation  connecting  x  and  y  he* 

a^      b'' 
Then  1^  =  ^  +  ^-1  =  0. 


du      2  X       du      2  y       dy  _       a^  P     x 

dx       a^  '     dy       6^        dx  2y  (^     y 

II.    Let  xy  =  C. 

Then  u  =  xy  —C  =  0. 

du _    ^    du _    .    ^y _ __y ^ 

dx     ^     dy         ^     dx  x 

Whenever  it  is  convenient  to  use  the  given  relation  to 
simplify  our  result,  we  are  of  coarse  at  liberty  to  do  so.  In 
this  case  we  can  readily  solve  the  given  relation  for  y, 
obtaining 


-f 

and  differentiating              -^  = -• 

dx          x^ 

III.    Let               u  =  x^  -{-  y^  —  axy  - 

=  0. 

dx                             dy 

^y  ^ 

dx 

3:r2- 
3/- 

-  ay 

-  ax 

*  It  was  necessary  above  to  emphasize  the  fact  that  we  were  dealing  with 
identities  in  order  that  the  correctness  of  the  step  by  which  we  deduced  (3) 
might  be  clear.  The  need  for  special  emphasis  of  the  character  of  our  equa- 
tions being  past,  we  return  to  the  use  of  the  ordinary  sign  of  equality,  though 
we  still  continue  to  deal  with  identities. 


298  CALCULUS  [Ch.  IX. 

EXERCISES   XXXI 

Find  — ^,  y  being  given  as  an. implicit  function  of  x  in  the  following 
dx 
relations : 

1.  xy'^  -\-x'^y  +  l=0.  '5.  (x2  -  y'^){ax  +  &)  =  0. 

2.  a2x2  _  52^2  _  ^2^,2  ^  0.  6.  X  sin  y  =  2/ sin  a:. 

3.  a:3-6a:2^  +  2a;2/2  + 73/8  =  0.  "^^  si"^  =  *^^^f* 

4.  a;^  —  y^  —  xy  =  0.  8.  a:^  =  3/==. 

9.    (x^  +  y^-  axy  -  b%x^  +  if)  =  0. 

Art.  11.  Homogeneous  functions.  A  function  of  two 
variables  is  said  to  be  homogeneous  of  degree  n,  if  the 
function  which  results  from  multiplying  each  variable  in 
it  by  the  same  constant  factor  e  is  c'^  times  the  original 
function.  In  symbols,  /(a;,  y')  is  homogeneous  of  the  ^th 
degree  in  x  and  y,  if 

(1)  f(cx,  cy^  =  ef(x,  y'), 

A  similar  definition  applies  to  functions  of  any  number 
of  variables.  Thus,  f{x^  y,  z^  iv)  is  homogeneous  of  the  nth 
degree  in  x^  y,  z,  w,  if 

(2)  fi^cx,  cy,  cz,  ew}  =  c'^f(x,  y,  z,  w}. 

EXAMPLES 

1.  ax  +  %  is  homogeneous  of  the  first  degree  in  x  and  y. 

2.  — ^-^I — ^  is  a  homogeneous  function  of  degree  —  1,  in  a:  and  y. 
3  x'^  +  9  xy 

3.  sin  ( ^   ~  y  )  is  a  homogeneous  function  of  degree  zero,  in  x,  y,  z,  w. 

4.  V^  +  yfyz  is  homogeneous  of  degree  f  in  x,  y,  z. 

5.  sin ^  is  not  homogeneous,  since 

(cy^  —(cx^  c{y^  —  x^) 

is  not  equal  to  the  original  function  multiplied  by  some  power  of  c. 


10-12.]  FUNCTIONS   OF  SEVERAL    VARIABLES  299 

EXERCISES   XXXII 

Determine  whether  or  not  the  following  functions  are  homogeneous ; 
and,  if  homogeneous,  of  what  degree? 

1 .  3  x^t/  —  5  x^y^  +  7  xy^  +  6  ?/*. 

2.  12  x^yz^  -  3  xY  +  -^  +  5  x^z. 

3.  5x7  +  2?/7  -3xY+  1. 
^  S  x\f/  +  11  x^yz^       • 

5^/622-2x3  +  4^8' 

5^    5  a:  +  2  //  +  .r^ 
2  X  —  3  ^  —  ^2 

^  5a: 


2  ?/3  —  3  2;^ 


8. 

9. 

.         .           x^ 
log  sm  -^ 

x^y  +  2/3 

10. 

tan  ^  +  -^^ 

11. 

arc  sm         ^  -^   • 

12. 

1 

</x-'  -  2/2 

13. 

V^+2/V^+^. 
Va: 

14. 

V2  a:5  +  V2  ^4^/  -  V2  x2//2. 

7.    los-. 


Art.  12.   Euler's  theorem  of  homogeneous  functions.     The 

partial  derivatives  of  a  homogeneous  function  possess  a 
remarkable  property  discovered  by  Euler,*  viz.  :  The  sum 
of  the  products  formed  by  multiplying  each  partial  derivative 
of  a  homogeneous  function  by  the  variable  with  respect  to  which 
the  derivative  is  taken.,  is  equal  to  the  original  function  multi- 
plied by  its  degree. 


*  Leonhard  Euler  (1707-1783)  was  the  son  of  a  clergyman  who  was  him- 
self a  pupil  of  James  Bernoulli  (p.  147).  Leonhard  Euler  was  a  pupil  of 
John  Bernoulli,  and  Nicholas  the  second  and  Daniel  Bernoulli  were  among 
his  fellow  students.  Having  attained  his  master's  degree  at  the  age  of  six- 
teen, Euler  was  soon  called  to  membership  in  the  Academy  at  St.  Petersburg 
(founded  1724).  He  accepted  the  invitation  in  1727,  and  resigned  in  1741 
to  become  a  member  of  the  Berlin  Academy,  then  flourishing  under  the 
fostering  care  of  Frederick  the  Great.  He  returned  to  St.  Petersburg  in  1766, 
and  though  he  became  blind  in  the  same  year,  he  continued  his  scientific 
activity  up  to  the  time  of  his  death.  Euler  was  a  mathematician  of  the  first 
rank,  and  contributed  largely  to  the  development  of  the  calculus  in  the 
eighteenth  century. 


300  CALCULUS  [Ch.  IX. 

We  give  the  proof  for  a  homogeneous  function  of  two 
variables,  but  the  method  may  be  used  equally  well  for  any 
number  of  variables. 

Let  u  =f(x,  y) 

be  a  homogeneous  function  of  degree  n  in  x  and  y.     Then 
Euler's  Theorem  asserts  that 

(1)  x---{-y  —  =nu. 

dx         by 

By  the  definition  of  homogeneous  functions, 

(2)  f{cx,  cy-)  =  e-f(x,  y}. 

Differentiating  this  identity  with  respect  to  (?,  we  have 
(p.  291), 

(3)  ^  d^-^^^^'  ^y^  "^  y  j-fC^^y  <^y)  =  nCfix,  y). 
Putting  c  =  1  in  the  identity  (3),  we  have 

(4)  X  ^A^'  !/)  +  !/  T-/(^'  ^)  =  ^/(^'  2/)- 

ox  dy 

or, 

(5)  a^^^+yp  =  nu. 

EXERCISES    XXXIII 

Verify  Euler's  Theorem  of  homogeneous  functions  for  the  following 
functions : 


4.  e      xy+y2 

5.  arc  tan 


1. 

5  x^y^  -  2  xK 

2 

^'    ^..  1    ^V^ 

y     '-^     x'  +  y^ 

3. 

^G  ,   2.y  ,        xy^      . 

y/x      Vx^  -  5  2/5 

x  +  2/ 


6.  iog£i+i£y. 

2/^  —  3  a:^^ 
7.    c^x^iy^i  +  c^x'^^y^^  +  c^x^^y^^  +  c^x'^iy^  where 

«!  +  &1  =  flg  +  ^2  =  ^3  +  ^3  =  ^4  +  ^4  =  ^* 


12-13. J         FUNCTIONS  OF  SEVERAL    VARIABLES 


301 


8.  Prove     Euler's     Theorem     for    homogeneous    functions    of    three 
variables. 

Verify  Euler's  Theorem  for  the  following  functions: 

9.  ax^y'^z  +  hxyz^  +  cy^  +  dxz^. 


11.  cot4/^-±l. 

^    X  +  z 


10.  sin 


X  +  y. 


12.   sin  —   +  log  ^  ^ 


i,  '6xy  —  z^ 

Art.  13.    The  focal  properties  of  the  parabola.     If  the 
equation  of  the  parabola  be  written  in  the  form  (p.  21) 

(1)        F(x,y)  =  f-2j?x  =  0, 
we  have 


(2) 
whence 


(3) 


dF        c,       ^F     ^ 

dx  dy 

dF 

dy  _      dx  _2p  _  p 
dx~~'dF~  2y~  y' 
dy 


The  equation  of   the  tangent  at  ^'^'  ^^* 

the   point  P    (Fig.  58),  whose  coordinates  are  a;^  and  y^. 
must  be  of  the  form  (p.  32) 


(4) 


^  -  ^1  =  (^  -  ^i)  tan  T. 


Substituting  for  tan  r  its  value  from  equation  (3),  and 
clearing  of  fractions,  we  have 


2  _ 


and  finally  by  (1)  the  equation  of  the  tangent  to  the  parabola 
becomes 


(5) 


yi/i  =  Pi^  +  a^i) 


302  CALCULUS  [Ch.  IX. 

It  is  apparent  from  (3)  that  the  tangent  at  0  is  perpen- 
dicular to  the  :r-axis.  The  point  0  is  called  the  vertex  of 
the  parabola. 

The  point  of  intersection  with  the  ic-axis  of  the  tangent 
to  the  parabola  at  the  point  P  is  found  by  putting  y  =  0 
in  (5);  then 

(6)  a;  +  ^1  =  0,    or   x=  —  x^^. 

If  T  be  this  point  of  intersection,  and  OQ  the  abscissa,  x^^ 
of  P,  equation  (6)  shows  that  OT  =  OQ. 

This  furnishes  a  very  simple  method  of  constructing  geometrically  the 
tangent  at  any  point  of  a  parabola.  We  drop  a  perpendicular  from  the 
point  upon  the  x-axis,  determine  a  point  equally  distant  from  the  vertex 
with  the  foot  of  this  perpendicular,  but  on  the  opposite  side ;  the  straight 
line  through  the  point  thus  determined  and  the  given  point  P  is  the 
tangent  desired. 

If  from  P  we  let  fall  a  perpendicular  upon  the  directrix 
of  the  parabola,  and  draw  PF  and  BT,  the  figure  PDTF 
is  easily  shown  to  be  a  rhombus  ;  for 

PD  =  PS  +  SD  =  x^  + 1, 

TF=TO+  OF=x^-\-^' 

Hence  PD  =  TF,  and  since  PD  and  TF  are  parallel,  and 
since  by  the  definition  of  a  parabola,  PD  =  PF,  the  figure 
is  a  rhombus.  The  diagonal  TP  bisects  the  angle  BPF. 
If  at  P  we  erect  a  perpendicular  PN  to  the  tangent  PT, 
then  PN  forms  equal  angles  with  PF  and  PP'.  PN  is 
called  the  normal  to  the  parabola  at  the  point  P.  If  we 
call  PF  the  focal  ray  of  the  point  P,  we  may  formulate  the 
theorem :    The   tangent   mid   the  normal  of  any  point  of  a 


13-14.]  FUNCTIONS  OF  SEVERAL    VARIABLES  303 

parabola  bisect  the  angles  formed  by  the  focal  ray  and  lines 
parallel  to  the  x-axis. 

This  property  of  a  parabola  is  important  in  optics.  For  when  rays 
of  light  parallel  to  the  principal  axis  of  a  parabola  impinge  upon  the 
latter,  and  are  reflected  from  it,  they  all  arrive  at  a  common  point  F\ 
for  instance,  PF  is  the  reflected  ray  of  the  impinging  ray  PP'.  This 
common  point  we  have  already  called  the  focus.  (Lat.  focus  =  fireplace.) 
Conversely,  if  a  source  of  light  be  at  F,  all  rays  coming  from  it  are 
reflected  so  as  to  be  parallel  to  the  principal  axis  of  the  parabola.  This 
is  also  true  when  the  parabola  is  rotated  around  its  principal  axis  so 
that  the  reflecting  line  becomes  a  reflecting  surface.  For  this  reason, 
concave  mirrors  which  are  intended  to  reflect  light  to  a  great  distance 
should  be  given  the  parabolic  form.  Hertz,  in  the  first  of  his  celebrated 
experiments  on  the  propagation  of  electric  rays,  made  use  of  thi^  property 
of  parabolic  surfaces.  He  employed  large  reflectors  of  sheet  zinc  bent 
into  the  form  of  parabolic  cylinders,  in  whose  focal  line  the  transmitter 
and  the  receiver  of  the  electric  waves  was  placed.  The  electric  rays 
passed  from  the  transmitter  to  the  first  parabolic  reflector  were  there 
reflected  so  as  to  become  parallel,  and  were  then  reflected  from  the 
second  reflector  to  the  receiver  placed  at  its  focus.  Also  Marconi,  in  his 
experiments  on  wireless  telegraphy,  is  attempting  to  confine  the  propaga- 
tion of  the  electric  waves  to  one  direction  by  the  use  of  copper  parabolic 
reflectors. 

Since  the  diagonals  of  the  rhombus  PDTF  bisect  each  other,  and  are 
perpendicular  to  each  other,  it  follows  that  the  foot  of  the  perpendicular 
from  the  focus  to  any  tangent  of  the  parabola  is  the  point  of  intersection 
of  that  tangent  with  the  tangent  at  the  vertex. 

Art.  14.  The  focal  properties  of  the  ellipse.  If  the  equa- 
tion of  the  ellipse  be  considered  in  tlie  form 

(1) 

then 

(2) 

and  therefore, 

(3) 

21 


F(x,y^  = 

.  ^^  _,  y'^ 

'a''      P 

1  =  0, 

dF 

dx 

2x     dF 
a^       dy 

62 

dy^ 
dx 

b'^x 
a^y 

802  CALCULUS  [Ch.  IX. 

It  is  apparent  from  (3)  that  the  tangent  at  0  is  perpen- 
dicular to  the  x-^xis.  The  point  0  is  called  the  vertex  of 
the  parabola. 

The  point  of  intersection  with  the  a;-axis  of  the  tangent 
to  the  parabola  at  the  point  P  is  found  by  putting  «/  =  0 
in  (5);  then 

(6)  x-\- x^  =  0^    or   x=  —  Xy 

If  jT  be  this  point  of  intersection,  and  OQ  the  abscissa,  x-^^ 
of  P,  equation  (6)  shows  that  OT  =  OQ, 

This  furnishes  a  very  simple  method  of  constructing  geometrically  the 
tangent  at  any  point  of  a  parabola.  We  drop  a  perpendicular  from  the 
point  upon  the  x-axis,  determine  a  point  equally  distant  from  the  vertex 
with  the  foot  of  this  perpendicular,  but  on  the  opposite  side ;  the  straight 
line  through  the  point  thus  determined  and  the  given  point  P  is  the 
tangent  desired. 

If  from  P  we  let  fall  a  perpendicular  upon  the  directrix 
of  the  parabola,  and  draw  PF  and  BT,,  the  figure  PDTF 
is  easily  shown  to  be  a  rhombus  ;  for 

PD  =  PS+SD  =  x^  +  t 

TF=TO^OF=x^  +  ^' 

Hence  PD  =  TF,  and  since  PD  and  TF  are  parallel,  and 
since  by  the  definition  of  a  pjarabola,  PD  =  PP,  the  figure 
is  a  rhombus.  The  diagonal  TP  bisects  the  angle  DPF. 
If  at  P  we  erect  a  perpendicular  PJSf  to  the  tangent  PT, 
then  PN  forms  equal  angles  with  PF  and  PP' .  PN  is 
called  the  normal  to  the  parabola  at  the  point  P.  If  we 
call  PF  the  focal  ray  of  the  point  P,  we  may  formulate  the 
theorem :    The   tangent  arid  the  normal  of  any  point  of  a 


13-14.]  FUNCTIONS  OF  SEVERAL    VARIABLES  303 

parabola  bisect  the  angles  formed  by  the  focal  ray  and  lines 
parallel  to  the  x-axis. 

This  property  of  a  parabola  is  important  in  optics.  For  when  rays 
of  light  parallel  to  the  principal  axis  of  a  parabola  impinge  upon  the 
latter,  and  are  reflected  from  it,  they  all  arrive  at  a  common  point  F; 
for  instance,  PF  is  the  reflected  ray  of  the  impinging  ray  PP'.  This 
common  point  we  have  already  called  the  focus.  (Lat.  focus  =  fireplace.) 
Conversely,  if  a  source  of  light  be  at  F,  all  rays  coming  from  it  are 
reflected  so  as  to  be  parallel  to  the  principal  axis  of  the  parabola.  This 
is  also  true  when  the  parabola  is  rotated  around  its  principal  axis  so 
that  the  reflecting  line  becomes  a  reflecting  surface.  For  this  reason, 
concave  mirrors  which  are  intended  to  reflect  light  to  a  great  distance 
should  be  given  the  parabolic  form.  Hertz,  in  the  first  of  his  celebrated 
experiments  on  the  propagation  of  electric  rays,  made  use  of  thi^  property 
of  parabolic  surfaces.  He  employed  large  reflectors  of  sheet  zinc  bent 
into  the  form  of  parabolic  cylinders,  in  whose  focal  line  the  transmitter 
and  the  receiver  of  the  electric  waves  was  placed.  The  electric  rays 
passed  from  the  transmitter  to  the  first  parabolic  reflector  were  there 
reflected  so  as  to  become  parallel,  and  were  then  reflected  from  the 
second  reflector  to  the  receiver  placed  at  its  focus.  Also  Marconi,  in  his 
experiments  on  wireless  telegraphy,  is  attempting  to  confine  the  propaga- 
tion of  the  electric  waves  to  one  direction  by  the  use  of  copper  parabolic 
reflectors. 

Since  the  diagonals  of  the  rhombus  PDTF  bisect  each  other,  and  are 
perpendicular  to  each  other,  it  follows  that  the  foot  of  the  perpendicular 
from  the  focus  to  any  tangent  of  the  parabola  is  the  point  of  intersection 
of  that  tangent  with  the  tangent  at  the  vertex. 

Art.  14.  The  focal  properties  of  the  ellipse.  If  the  equa- 
tion of  the  ellipse  be  considered  in  the  form 

(1)  l'(.,2,)=g  +  g-l  =  0, 

then 

(2) 

and  therefore, 

(3) 

21 


dF_^2x^    dF  ^2y^ 
dx        a^       dy        6^ 

dy  _  _  h'^x 
dx  a^y 


306  CALCULUS  [Ch.  IX. 

easily  that  the  normal  bisects  the  angle  made  by  the  focal 
rays  ;  i.e.^ 

At  every  point  of  an  ellipse  the  tangent  and  the  normal  bisect,, 
respectively,,  the  exterior  and  the  interior  angle  formed  by  the 
focal  rays. 

If  we  imagine  a  source  of  light  to  be  placed  at  F^,  then  PF^  repre- 
sents the  path  of  a  ray  of  light  emanating  from  that  point-source.  If 
the  ellipse  be  a  reflecting  line,  the  reflected  ray  must  make  the  same 
angle  with  the  normal  as  the  incident  ray,  and  since 

F^PN  =  NPF^, 
PF2  is  the  reflected  ray ;  i.e., 

All  rays  of  light  having  as  source  one  of  the  foci  of  an  ellipse  are  refected 
so  as  to  meet  in  the  other  focus. 

This  is  true  of  all  rays  which  travel  in  straight  lines  and  which  are 
reflected  according  to  the  law  that  the  angle  of  reflection  is  equal  to 
the  angle  of  incidence.  This  property  of  the  ellipse  is  the  cause  of  the 
peculiar  echo  and  concentration  of  sound  in  certain  arched  halls,  grottos, 
etc.  It  may  be  also  illustrated  by  the  well-known  experiment  of  placing 
an  easily  inflammable  substance  in  one  focus  of  an  elliptic  surface  and 
igniting  it  by  means  of  a  glowing  coal  placed  in  the  other. 


Art.  15.   The  asymptotes  of  the  hyperbola.     Writing  the 
equation  of  the  hyperbola  in  the  form 


(1) 

^(-^^)  =  ^- 

we  have 

(2) 

BF      2x 

dx       a^' 

and  . 

(3) 

dy  _  b^x 
dx      a^y 

6^ 


2x    ^^_2y 
'    dy  52 ' 


14-15.]  FtlNCTiONS   OF  SEVERAL    VAttlABLES  S07 

We  find  the  equation  of  the  tangent  at  the  point  (a:^,  ^j) 

to  be 

(J  u  b  jT 

^  _  M  -  ^  _  ^ . 

^^  a2        52  -  ^2       ^2  ' 

whence  from  (1), 

^^;  ^2     ^,2  -^• 

The  abscissa  of  the  point  of  intersection  of  the  tangent 
with  the  a;-axis  has  the  value 

(5)  x^"-. 


Of  particular  interest  are  the  asymptotes,  already  illus- 
trated (p.  59)  and  now  defined  as  the  limiting  positions 
whicli  the  tangents  approach  as  the  points  of  contact  move 
out  on  the  hyperbola  beyond  all  bounds.  Because  of  the 
symmetry  of  the  hyperbola  it  is  necessary  to  consider  it  in 
one  quadrant  only,  the  first,  for  instance ;  as  the  point  of 
contact  moves  out  beyond  all  bounds,  its  abscissa  likewise 
increases  beyond  all  bounds,  and  we  have 

lim    ^  f\ 

^  =  ^  •  ^-    or  a:=0; 

that  is  to  say,  the  asymptote  passes  through  the  origin.  To 
find  from  equation  (3)  the  angle  which  the  tangent  anakes 
with  the  a;-axis,  we  must  ascertain  the  limit  of  the  ratio  of  x 
to  y  when  x  grows  beyond  all  bounds.     By  equation  (1), 


808  CALCULUS  [Ch.  IX. 

Extracting  the  square  root,  and  taking  the  limit, 

...  lim  I  _   lim  [^  __  ^Y  ^  f^Y  =  +  ^ 

If  we  substitute  this  value  in  equation  (3)  and  denote  the 
angle  in  question  by  </>,  we  have 

(7)  tan</)  =  ±-.  -  =  ±-. 

As  we  are  restricting  our  considerations  to  the  portion  of 

the  hyperbola  in  the  first  quadrant,  the  positive  sign  is  to  be 

taken. 

If  we  now  construct  a  rectangle  with  its  sides  parallel  to 

the  axes,  and  cutting  off  on  the  axes  the  distances  (Fig.  26, 

p.  55), 

OA^  =  OA^  =  a 

and  .  OB^  =  OB^  =  b, 

respectively,  the  diagonal  in  the  first  quadrant  is  the  required 
asymptote.  Because  of  the  symmetry  of  the  hyperbola  it 
follows  that  this  straight  line  produced  is  the  asymptote 
to  that  part  of  the  hyperbola  which  lies  in  the  third  quad- 
rant, and  that  the  other  diagonal  of  the  rectangle  is  likewise 
an  asymptote  to  the  hyperbola  in  the  second  and  in  the 
fourth  quadrant. 

We  have  thus  reached  the  same  results  for  any  hyperbola 
which  we  have  previously  found  for  the  equilateral  hyperbola 
(p.  61). 

Observing  that  the  second  diagonal  makes  the  angle  tt  — <^ 
with  the  a;-axis,  and  tan  </>  =  —  tan  (tt  —  <^),  we  can  write 
the   equations  of  the   asymptotes   at  once  from   the  slopes 


15.]  FUNCTIONS   OF  SEVERAL    VARIABLES  309 

(Eq.  7),  and  from  the  fact  that  the  asymptote  passes  through 
the  origin,  viz. 

y  =  -x  orf- =  0, 

a  ha 

and 

y  = X  orf-  +  -  =  0. 

a  ha 


CHAPTER  X 
INFINITE   SERIES 

Art.  1.  Definition.  A  sequence  of  terms  which  are 
formed  according  to  some  rule  or  law,  so  that  more  terms 
can  be  written  according  to  the  same  rule  or  law,  is  called  a 
series  of  terms,  or  a  series.  « 

For  example, 

1,     2,     3,     4,     5,     6,    *..., 

is  a  series,  the  law  of  which  is  that  each  term  is  greater  by 
unity  than  the   one  preceding  it.     We   could   extend   the 
series  by  the  terms  7,  8,  9,  10,  •••,  as  far  as  we  like. 
.    The  following  are  further  examples  of  series : 

1.  1,     3,     9,     27,     ..., 

2.  1,     4,     9,     16,     ..-, 

'*•     -*-'      "5'       2  5'       12^'  •> 

4.  1,  3,  5,  7,     9,     ..., 

5.  1,  4,  5,  8,     9,     12,     13,     16,     17,     -.., 

6.  10,     8,     6,  4,     ..., 

7.  1,  2,  3,  5,     8,     13,     21,     34,     55,     -.. 

Exercise.  Discover  the  law  of  each  series  by  inspection,  and  write  the 
next  four  terms  of  each. 

If  the  law  of  the  series  is  such  that  there  is  no  bound  to  the 
number  of  terms  which  may  be  written,  it  is  called  an  infinite 
series.     A  sequence  of  terms  is  not  called  a  series  when  no 

310 


1-2.]  INFINITE  SERIES  Sll 

law  can  be  discovered  according  to  which  additional  terms 
can  be  written. 

Art.  2.  The  sum  of  infinite  series.  The  fraction  ^  may- 
be converted  into  the  decimal  fraction  0.3333  ••• ;  i.e,  into 

__3_    I    _.3 _    I 3 I 3. |_  . . . 

10  ^  100  ^   1000   ^   10000  ^ 

We  say,  ordinarily,  that  this  decimal  which  never  terminates 
(or  the  unbounded  series  of  terms  to  which  it  is  equivalent) 
is  equal  to  1.  This  is  not  strictly  true,  but  rather,  ^  is  the 
limit  which  the  decimal  (or  the  series  of  terms)  approaches 
as  it  is  carried  out  farther  and  farther. 

For,  -^^  is  not  -J;  3%  +  -^f  o'  ^^  '10%  ^^  ^^^  h  ^^^  ^^  differs 
less  from  i  than  j\  does  ;  ^%  4-  yf  0  +  i  oW'  or  -f^%%  is  not  1 
but  it  differs  less  from  ^  than  either  -^^  or  -^q\  ;  and  thus  by 
taking  more  terms  we  may  reach  a  sum  which  shall  differ  as 
little  from  ^  as  we  like;  i.e.  the  sum  of  the  terms  of  the 
series,  as  we  take  more  and  more  of  them,  approaches  the 
limit  ^.     Similarly,  the  series 

2   +  4  +   8  +  A  +  32   +  •*• 

has  unity  as  the  limit  of  the  sum  of  its  terms. 

The  sum  of  the  ^terms  of  a  series  as  more  and  more  are 
taken  may  not  approach  any  limit.  This  is  the  case,  for 
instance,  in  the  series 

1,  2,  3,  4,  5,  .-.. 

Here  evidently  the  sum  grows  large  beyond  all  limits  as 
more  and  more  terms  are  included  in  it. 

If  the  sum  of  the  first  terms  of  a  series  approaches  some 
definite  limit  as  more  and  more  terms  are  taken.,  that  limit  is 
defined  to  be  the  sum  of  the  series. 


312  CALCULUS  [Cii.  X. 

We  often  write,  accordingly, 

1—     3i3i  3        4_... 

3  —  TO  ^   1^0  ^  TOGO"  ^        ' 

1  =  2   +  4  +   8  +  16   +  "*  ' 

but  these  are  simply  abbreviations  for  the  following : 

1  _  ]irv^  5    3__     I     __3 I     ...i 

A  series  which  has  a  sum  is  called  convergent.  If  the  sum 
of  the  first  terms  of  a  series  can  be  made  as  large  as  we 
please  by  taking  enough  terms,  the  series  is  divergent. 

We  have  given  one  example  of  a  divergent  series  above , 
we  now  add  another,  viz,^ 

At  first  glance  it  may  seem  as  if  this  series  should  be  con- 
vergent, but  it  may  be  proved  to  be  divergent  as  follows : 
We  compare  the  two  series 

(1)     l+|  +  i  +  i  +  i  +  -J  +  KI  +  i  +  -+TV  +  -' 

(2)  i  +  Ki  +  i  +  i  +  4  +  i  +  i  +  A  +  -  +  iV  +  -> 

of  which  the  first  is  the  given  series,  %nd  the  second  lias 
every  one  of  its  terms  equal  to  or  less  than  the  correspond- 
ing term  of  the  first.  The  sum  of  any  number  of  terms  of 
(1)  will  therefore  be  greater  than  the  sum  of  the  same  num- 
ber of  terms  of  (2),  and  if  we  can  show  that  a  sum,  large  at 
will,  can  be  obtained  by  taking  enough  of  the  terms  of  (2), 
we  shall  thereby  have  shown  that  the  same  can  be  done  by 
taking  enough  of  the  terms  of  (1). 
If  we  write  (2)  thus, 

(3)  J  +  J  +a  +  D  +  (i  +  i  +  i  +  i)+(iV  +  -  +  iV)+  •••' 


2-3.]  INFINITE  SERIES  313 

and  add  the  terms  within  the  parentheses,  we  have  (2)  in 

the  form 

1  4_  1  4_  1  4_  1  _|_  1  4_  ... 

By  taking  enough  terms  of  this  we  clearly  can  obtain  a 
sum  large  at  will,  and  hence  (1)  is  divergent ;  it  is  called 
the  harmonic  series. 

The  series 

(4)  l-i  +  i-i  +  i-i+-, 

which  differs  from  the  harmonic  series  only  in  having  its 
signs  alternately  plus  and  minus,  may  be  put  into  the  two 
forms 

(5)  (l_a)  +  Q-^)  +  (i_i)+..., 

(6)  and  1- (1-1) -(1-1) -(1-1)..., 

in  each  of  which  the  quantities  in  parentheses  are  positive. 
It  is  easily  seen  that  in  the  first  form  the  sum  of  any  number 
of  terms  of  the  series  is  greater  than  the  first  term  (1  —  |-), 
and  in  the  second  that  it  is  less  than  1 ;  it  lies,  therefore, 
between  1  and  |. 

Art.  3.  The  geometric  series.  The  simplest  example  of  a 
convergent  series  is  the  geometric  series 

(1)  1  +  «  +  «2  _|_  ^3  _|_  ^^4  ^ ^ 

for  which  we  have  the  equation  * 


(2)*    •   l  +  «-f-«2...^«. 


.-1  _!-«"_     1 


1  -  «       1  -  a      1 


If  we  suppose  «  to  be  a  proper  fraction  and  n  to  increase 
without   bound,  then  the   first   fraction  of   the   right-hand 

*  Formula  52,  Appendix. 


^14  CALCULUS  [Ch.  X. 

member  remains   unchanged,  while  the  second  approaches 
zero.     We  obtain,  therefore, 

(3)  limn +«  +  a2  +  «3  +  ...J  =_!_. 

1  —  « 

TTie  unlimited  geometric  series  (1),  in  which  a  is  less  than  1, 
is  convergent^  and  its  sum  is 


1-a 

The  practical  applicability  of  infinite  series  depends  upon 

the   rapidity   of    their   convergence ;     or,    in   other   words, 

upon  how  many  terms  we  must  take  in  order  to  obtain  a 

sum  which  shall  differ  as  little  as  we  wish  from  the  limit. 

The  most  favorable  case  occurs  when  even   two  or  three 

terms  give  a  very  close  approximation  to  the  limit.     Taking 

a  simple  numerical  example,  suppose  we  wish  to  find  the 

0  432 
value  in  decimal  notation  of  the  fraction  — .     We  put 

0.998  ^ 

^•432  =  0.432         1 


0.998  1-0.002 

=  0.43211  -f  0.002  +(0.002)2+  ... I, 

and  since  0.0022  =  0.000004,  the  expression  in  which  only 
the  first  two  terms  of  the  series  are  used  gives  correctly  the 
first  five  decimal  places  of  the  value  sought.  If  a  greater 
accuracy  be  required,  as  to  the  seventh  or  eighth  decimal 
place,  three  terms  of  the  series  suffice ;  for 

^^  =  0.432(1  +  0.002  +  0.000004) 

0.998 

=  0.432-1.002004; 

the  third  term  is  used  as  a  correction,  and  each  following 
term,  if  used,  has  a  similar  effect. 

Art.  4.   General  theorems  on  the  convergence  of  series. 
Series  with  alternating  signs.     Before  any  use  is  made  of  a 


3-4.]  INFINITE  SERIES  315 

series  occurring  in  a  mathematical  investigation,  it  must 
first  of  all  be  decided  whether  or  not  the  series  is  conver- 
gent. This  is  often  a  problem  of  great  difficulty.  We  can 
consider  only  the  simplest  cases. 

From  now  on  we  shall  designate  the  terms  of  any  given 
series  by 

^1^       ^2'       ^^3'       ^4'       "*' 

the  subscript  indicating  the  position  of  a  term  in  the  series, 
a^„,  for  instance,  being  the  mth  term. 

We  denote  the  sum  of  the  first  n  terms  of  the  series  by  jS^, 
Accordingly, 

^3  =  ^1  +  ^2  +  ^3^ 


>^^  =  «1  +  «2  +  ^3  -I ^  ^k-l  +  (^lo 


aS;  =  «i  +  «2  +  «3  H V  a^^i  +  a^. 


With  this  notation,  and  denoting  the  sum  of  a  convergent 
series  by  aS',  the  definition  of  the  sum  becomes 

cr        lim    a 
n  =  GO     " 

If  a^  is  the  rth  term  of  our  series,  we  need  to  consider,  in 
order  to  determine  whether  the  series  converges,  only  the 
terms  from  a^  on ;  that  is,  the  terms 

(1)  a^  +  a^+i  4-<^r+2+  •••; 

since  the  sum  of  the  original  series  is  equal  to  the  sum  of  (1) 
plus  the  sum  of  the  other  r  —  1  terms,  ^^  +  ^g  +  •••  +  a^^i. 


316  CALCULUS  [Ch.  X. 

A  series  with  alternating  signs  can  be  represented  by 

(2)  a^  -  ^2  +  ^3  -  ^4  +  ^5  -  ^6  +  •••' 

wherein  a^,  a^,  a^^  ••.,  are  positive  numbers.     We  shall  now 
prove  the  following  theorem  : 

If  the  terms  of  an  infinite  series  with  alternating  signs  con- 
tinually decrease  numerically/  and  approach  the  limit  zero,  the 
series  is  convergent. 

The  proof  is  similar  to  that  given  in  the  example  (p.  312); 
we  write  the  series  in  the  forms : 

(3)  (^1  -  a^)  +  (^3  -  «4)  +  (^5  -  ag)  +  ..., 

(4)  a^-  (^2  -  a^)  -  («4  -  ^5)  -  K  -  ^7) • 

The'  differences  in  the  parentheses  are  all  positive,  inas- 
much as  we  assumed  the  terms  to  decrease  continually.  It 
follows,  then,  from  the  first  form,  that  the  sum  of  any  num- 
ber of  terms  of  the  series  is  greater  than  a^  —  a^',  and  grows 
larger  as  more  terms  are  taken ;  and  from  the  second  form, 
that  it  is  smaller  than  a^,  and  grows  smaller  as  more  terms 
are  taken.  Therefore,  the  sum  of  any  number  of  terms  of 
the  series  (3)  always  increases  as  more  terms  are  taken,  and 
is  always  greater  than  a^  —  ^21  ^ut  less  than  a^ 

It  is  readily  seen,  further,  that  if  a  variable  quantity  con- 
tinually increases  hut  always  remains  less  than  some  constant, 
a,  the  variable  approaches  some  limit.  For,  either  the  variable 
may  be  made  to  differ  little  at  will  from  a,  in  which  case  a 
is  the  limit,  or  the  difference  between  the  variable  and  a  must 
always  be  at  least  equal  to  a  certain  quantity,  d,  say.  In 
this  case  the  variable,  though  always  increasing,  can  never 
exceed  a  —  d.     If  it  approaches  near  at  will  to  a  —  d,  the 


4.]  INFINITE  SERIES 

latter  is  the  limit.  If  not,  the  same  reasoning  can  be 
applied  to  a  —  d  which  we  have  just  applied  to  a.  By  thus 
repeatedly  diminishing  the  quantity  which  the  ever  increas- 
ing variable  can  never  exceed,  we  see  that  there  must  exist 
a  value  which  the  variable  can  never  exceed,  but  still  to 
which  it  may  approach  near  at  will.  That  is,  the  variable, 
while  ever  increasing,  approaches  a  limit. 

Applying  this  to  the  series  (3),  we  see  that  the  sum  of 
more  and  more  terms  approaches  a  limit  (lying  between 
a  J  —  ^2  ^nd  a^),  i.e.  the  series  (3)  is  convergent. 

Similarly,  the  series  (4)  may  be  shown  to  approach  a 
limit  lying  between  a^  —  a^  and  a^ 

We  show,  finally,  that  (3)  and  (4)  approach  the  same 
limit.  The  nth.  term  of  (3)  is  a^n-i  —  «2«^  ^nd  the  nih.  term 
of  (4)  is  —  {a2n-2  —  chn-i)'  The  difference  between  the  two. 
sums  of  n  terms  from  each  series  is  a-zn',  and  this  term  bv 
hypothesis  approaches  zero  as  n  grows  beyond  all  bounds 
Calling  the  sums  aS'^  and  aS"„,  we  may  write 

and  taking  the  limit, 

lim  /S^„  =  lim  aS''^. 

EXERCISES  XXXIV 

In  the  following  series,  first  discover  the  law  by  inspection,  then  write 
several  terms  more,  then  the  nth.  term ;  decide  whether  or  not  the  series 
is  convergent,  and  if  so,  between  what  values  the  limit  must  lie. 

1.  8  -  4  +  2  -  1  +  1  -  1  +  i  -  ^1^  +  .... 

2.  a  -  a2  +  a3  -  «*  +  «5  _  ^^6  ^  ....  («  <  1). 

3.  2-l  +  |-^  +  i-i  +  i-^+  .... 

aa  +  1      a  +  2      a  +  3     a  +  4 


318  CALCULUS  [Ch.  X. 

fi       n  1  ,  1  1,1 


a-^k      a^+2k      a^-\-'dk      d'+^k 
Q        a  a  +  1   ,  «  +  2      g  +  3  , 

~r T 7  i-  •••• 


a+1      «+2      a+8      a+4 

a         a+la+2a+3 

10.  (a  +  x)  -(a2  +  a;2)  +  (a^  +  x^)  -  (aH  x^)  +  ....  (a  <  1 ;  x  <  1 ). 

11.  2-t  +  Y-ff  +  ff--MI+-.  13.    a_^%^-^+.... 

2       3       4 

1.2      3.4      5.6     7-8  ^^'  2!      4!~6!         * 

Hint:     _L-  =  i-l,etc.  15.   :. -^  + ^ -^  +  .- 

5-6      5     6  3!     5!     7! 

Art.  5.  Series  with  varying  signs.  In  case  there  is  only 
a  limited  number  of  terms  having  one  of  the  signs  (for 
instance,  if  there  are  only  r  negative  terms),  then  there  will 
be  a  last  one  of  these  terms,  and  from  this  term  on  all  the 
terms  will  be  of  like  sign,  and  the  convergency  will  be  deter- 
mined under  the  rules  for  series  all  of  whose  terms  are  of 
like  sign.  (In  the  instance  given,  if  a„  be  the  rth  (and  last) 
negative  term,  the  series  will  be  convergent  or  not,  according 
as  a„+i  +  a„^2  +  *•*?  with  all  positive  terms,  is  convergent  or 
not.) 

In  case  there  is  a  boundless  number  of  terms  of  either 
sign,  the  following  theorem  will  enable  us,  in  this  case  also, 
frequently  to  determine  the  convergency  by  means  of  the 
theorems  for  series  of  like  sign. 

A  series  with  varying  signs  is  convergent  if  the  series 
deduced  from  it  hy  making  all  signs  positive  is  convergent. 


4-5.]  INFINITE  SERIES  319 

By  hypothesis,  there  is  a  boundless  number  both  of  posi- 
tive and  of  negative  terms,  and  the  series  resulting  from 
making  all  signs  positive  is  convergent.  We  now  consider 
two  infinite  series,  one  made  up  of  the  positive  terms  of  the 
given  series,  the  other  of  its  negative  terms  taken  positively. 
Both  of  these  series  must  be  convergent,  since  each  is  either 
convergent  or  grows  beyond  all  bounds ;  and  if  the  latter, 
the  original  series  would  also  increase  beyond  all  bounds 
when  its  terms  are  all  taken  positively. 

Let  L^  and  Zg  ^^  ^^^  limits  of  these  series,  and  T^  and  U„ 
the  sums  of  their  first  n  terms  respectively,  and  let  tS^  denote, 
as  usual,  the  sum  of  the  first  n  terms  of  the  given  series. 
Then  r.       ^       .. 

and  both  q  and  r  will  grow  large  without  bound  as  n  does  so. 

But  T^  =  L^+e^, 

and  U,.  =  L^-\-  e^, 

where  e^  and  e^  approach  zero  as  q  and  r  increase  without 
bound. 

Hence,  8^  =  L^  — L^-\- e^—Cr^ 

and  „!]:"„  S„  =  L,-  i„ 

or  the  given  series  is  convergent. 

The  theorem  just  proved  states  a  condition  which  is  suffi- 
cient but  not  necessary  for  the  convergence  of  series  with 
varying  signs.  For  instance,  we  have  proved  (p.  313)  that 
the  series 

is  convergent,  and  (p.  312)  that  the  series 

1  +  i  +  i  +  i  +  i  +  K-    . 

is  divergent, 

22 


320  CALCULUS  [Ch.  X. 

Akt.  6.   Series  whose   signs   are   all   positive.     We   now 

turn  to  series  all  of  whose  terms  are  of  like  signs,  which, 
without  loss  of  generality,  we  may  assume  to  be  positive. 
We  may  represent  such  a  series  by 

(1)  «i  +  ^2  +  ^3  +  ^4  +•••  +««  +  ««+!  +  ••> 


^ny  ^rt+1' 


where  we  consider  all  the  quantities  a^,  a^,  a^,  •• 
to  be  positive. 

In  this  case,  the  more  terms  we  take  the  larger  their  sum 
will  become.  To  prove  the  series  convergent,  it  is  necessary 
and  sufficient  to  show  that  no  matter  how  many  terms  we 
take,  their  sum  cannot  be  larger  than  some  definite  quantity 
(p.  316). 

This  may  often  be  done  by  the  following  theorem  : 

If  in  an  infinite  series  of  positive  terms  only^  the  quotient  of 
every  term  divided  by  that  which  precedes  it  is^from  a  certain 
term  on^  less  than  some  quantity  which  is  itself  less  than  unity, 
the  series  is  convergent. 

To  prove  this  theorem,  we  assume  that  all  the  quotients 

'  ^t+l        ^r+2        <^r+3 

-,       , 


^r  ^r+1         ^r+2 

are  less  than  some  quantity  q  which  is  itself  less  than  unity. 
We  have  then  the  following  relations : 

^r+l    ^  ^r+2  ^r+3    ^  ^r+4 


ir^^'       ^<^'       ^'^^^       a~~^^ 

<*r  "r+1  ^r+2  "r+3 

etc. 


Taking  the  first  relation  to  start  with,  and  then  the 
products,  first  of  the  first  two,  then  of  the  first  three,  then 
of  the  first  four,  etc.,  etc.,  we  find 


6.]  INFINITE  SERIES  321 

— -  <  q,     or    a^+i  <q  •  a^; 

-^<q%  or  a^+2<r -^r; 

-^<q%    or   a,.+3<^3.^^. 


~--<q\    or   a^+4  <  5^*  .  a^  ; 

and  by  adding 

(2)  a^+i-h  ar+2  +  a^+s  -\ <  g^r  +  9'X  +  ^a^  +  ••• ; 

and,  on  adding  a^,  to  each  member, 

(3)  a^  +  a,.^.i  +  a^+2  H <  «r(l  +  g  +  ^^  +  ^^  +  •••)• 

But  by  equation  (3)  (p.  314), 


hence 

(4)  '  a^  +  ^r+l  +  «r+2  +  -  <  zr^^' 

1-q 

Accordingly,  the  series  (3),  and  therefore  the  series  (1), 
is  convergent. 

Exercise.  Prove  similarly  that  if  in  any  series,  whose  terms  are  all 
positive,  the  ratio  of  each  term  to  the  preceding  is,  from  a  certain  term 
on,  greater  than  or  equal  to  unity,  this  series  is  divergent. 

We  mention  here  two  theorems  easily  seen  to  be  true, 
which  may  often  be  applied  with  advantage.     If   in   any 
given  series, 
(5)  «i  +  «2  +  ^3+ •••' 


322  CALCULUS  [Ch.  X. 

all  of  its  terms  after  a  certain  one  are  smaller  than  the 
corresponding  terms  of  a  series  known  to  be  convergent, 
then  the  series  (5)  is  also  convergent. 

If,  however,  all  the  terms  of  (5),  from  a  certain  one  on, 
are  greater  than  the  corresponding  terms  of  a  series  known 
to  be  divergent,  then  the  series  (5)  is  divergent. 

The  theorems  which  we  have  proved  will  usually  suffice 
to  determine  the  question  of  convergency  in  the  simple 
series  which  we  shall  take  up.     The  discussion  of  the  ratio 

-^—  is  especially  effective. 

Akt.  7.  Rapidity  of  convergency.  We  have  already 
stated  (p.  314)  that  the  practical  applicability  of  infinite 
series  depends  upon  their  rapidity  of  convergency ;  the 
greater  the  number  of  the  terms  which  must  be  considered, 
the  less,  of  course,  is  the  application  of  the  series  to  be 
recommended.  To  ascertain  how  many  terras  should  be 
taken  to  secure  any  desired  degree  of  approximation  to  the 
limit  of  the  series,  it  is  necessary  to  be  able  to  determine 
how  great  the  error  will  be  if  the  series  is  discontinued 
at  any  given  term ;  this  may  readily  be  done  in  the  cases 
of  the  series  we  have  hitherto  treated.  The  error  in  the 
geometric  series  is  given  at  once  by  equation  (2),  p.  313. 
Writing  that  equation  in  the  form 

(1)  -i-  =  1  -h  «  +  «2  +  ...  +  a-i  +  _^, 
1  —  a  \  —  a 

we  see  that  the  value  obtained  by  taking  the  first  n  terms 

will  differ  by from  the  limit  of  the  series.     On  sub- 

^  1-a 

stituting   —  13  for  «,  we  obtain 

(2)  ^=l-/3  +  /32-^3+-  +  (-l)'-»/3«-i  +  (-l)''j|J^- 


6-8.]  INFINITE  SERim  823 

It  now  follows  that  the  errors  made  by  taking  the  first 
n  terms  of  the  series  in  question,  instead  of  the  limits 

and 


1  -  a  1+  /3' 

are 

^a„d(-l)»^, 

respectively ;   (in  the  case  of  the  second  series,  the  sum  is 
alternately  too  large  and  too  small). 
For  the  series  (p.  320) 

ai  +  «2  +  ^3+  ••• 

it  follows,  similarly,  from   equation   (4),  p.   321,  that   the 
error  occasioned  by  putting  for  this  series  the  series 

is  less  than 

(3)  ^^ 


1-q 
where  5'  has  been  defined  (p.  320). 

Art.  8.  Application  to  the  series  for  e.  We  found 
(p.  138) 

1,1,1,1,1,         ,1^ 

1      2 !      3!      4 !  n\ 

Two  questions  arise  :  I.  Is  this  series  convergent  ? 
II.  What  is  the  error  committed  by  discontinuing  the  series 
at  any  given  term  ? 

Forming  the  ratio  of  two  consecutive  terms,  a^^  and  a^^^^ 
we  have  for  this  series, 

«m+i  _  1 


324  CALCULUS  [Ch.  X. 

Since,  from  the  third  term  on,  —  is  less  than  one-half,  the 

m 

series  is  convergent  according  to  p.  320.     If  we  express  its 
approximate  value  by  the  sum  of  the  first  m  terms  and  put 

9'  =  — ,  the  error  committed  is  less  than 


m 

^m+i m      _    1 

^_i  m  —  1      m  I  \       m 

m 


H 


In  particular,  a  value  obtained  by  taking  only  tlie  first 
seven  terms  into  consideration,  differs  from  the  true  value 
by  less  than 

7 !  V    ^6/     4320  ^^  ^     ' 

a  result  with  which  the  value  found  on  p.  138  agrees. 

EXERCISES   XXXV 

Determiue  whether  or  not  each  of  the  following  s6ries  is  convergent  :* 

1.  1  +  I  +  I  +  1  +  t\  +  ir\  +  -e'?  +-. 

2.  1+A  +  A  +  13  +  1Z+.... 

2!      3!     4!      5! 

3.  1  +  1  +  1  +  1  +  1+  .... 

22     33     44      55 

4.  l  +  a  +  ^'  +  l^-f^Vf +.-.whena<l. 

2       3       4       5 

5.  l  +  2!  +  33  +  4^^.5^  +  .... 

1      22      38  ^  44 

7.  1  +  1  +  J-+       1        ■  1  ■ 


3      3.4     3.4.5     3.4.5.( 

a  1  +  1  +  1  +  1  +  1  +  15  +  12  _, 

22      33      44      55      66      V 


*  The  determination  may  sometimes  be  made  in  more  ways  than  one  by 
applying  different  ones  of  the  previous  criteria. 


8-9.]  INFINITE  SERIES  325 

9.    1+i  +  f +  |  +  f  +  f +-. 

10.  A  +  JL  +  Jl  +  J-^.  .... 

2!      3!      4!      5! 

lyt  rj(*Jt  •T'O  '¥•4 

".   l+|  +  |  +  |  +  |j  +  ...when.<l. 

13.  x  +  25a;2  +  3^x3  +  45x4  +  55^5  +  G^x^  +  ...  when  a:  <  1. 

14.  1  +-^  +  -£_  +  _^  +  ,^-^+  ...  whena;<l. 

1  +  1      4  +  2      9  +  3      16  +  4 

15.  Consider  the  series  of  No.  14  when  x>\. 

Art.  9.  Maclaurin's  Theorem.  It  is  a  common  pro- 
cedure in  scientific  investigations  to  try  to  substitute  an 
approximate  formula,  obtained  empirically,  for  a  law  repre- 
senting the  unknown  course  of  ^a  process  of  nature.  Con- 
sidering, for  instance,  the  expansion  of  a  rod  that  at  the 
temperature  0°  C.  has  unit  length,  the  crudest  supposition 
which  we  can  make  is  that  the  length  of  the  rod  does  not 
vary  with  the  temperature.  This  supposition,  which  may 
be  expressed  by  the  formula 

(1)  ^  =  1, 

is  sufficiently  accurate  for  many  practical  purposes.  If  we 
make  the  supposition  that  the  expansion  is  proportional 
to  the  temperature,  we  can  represent  the  length  of  the  rod 
at  the  temperature  0  by  the  formula 

(2)  Z  =  l  +  «(9, 

in  which  a  is  the  coefficient  of  expansion.  The  formula  (^2) 
gives  a  closer  approximation  to  the  length  at  any  tempera- 
ture than  does  (1),  while  the  following  is  a  formula  corre- 
sponding still  more  closely  to  the  actual  length, 

(3)  i  =  \^ae^  ^e\ 


826  ,      CALCULTTS  [Ch.  X. 

V 

where  a  and  jS  are  constants,  which  may  be  determined  by 
comparing  the  formula  with  the  results  of  observation.  For 
example,  the  formula  for  the  linear  expansion  of  a  rod  of 
platinum  has  been  found  to  be 

1  =  1  +  0.00000851  e  +  0.0000000035  (92, 

where  6  is  the  temperature ;  it  is  apparent  that  the  term  ^6^ 
has  the  character  of  a  correction,  rendering  the  value  of  I 
more  nearly  exact ;  in  this  particular  case  the  correction  has 
so  slight  a  value  that  the  formula  is  accurate  enough  for  all 
practical  needs.  If  this  should  not  happen  to  be  the  case, 
we  could  proceed  a  step  farther  and  add  a  third  term,  as 
j6^  (a  correction  to  the  correction),  etc.,  etc. 

The  question  arises  whether  this  formula  could  not,  by 
the  proper  corrections,  be  made  absolutely  accurate.  It  is 
at  once  evident  that  to  attain  absolute  accuracy  we  must 
take  all  possible  corrections  into  account,  so  that  the 
formula  may  become  an  infinite  series,  as 

and  it  is  easily  seen  that  such  series  would  be  convergent 
from  the  very  nature  of  the  separate  terms. 

In  the  example  treated  above  it  was  required  to  find  a 
formula  giving  the  length  I  of  the  rod  for  any  value  what- 
ever of  the  temperature.  The  length  I  is,  therefore,  a  func- 
tion of  the  temperature  ^,  and  the  office  of  the  formula  is  to 
give  expression  to  this  unknown  function,  /(^).  If,  to 
treat  the  problem  generally,  we  denote  the  variable  upon 
which  the  function  depends  by  x  and  the  function  by  /(re), 
we  wish  to  express  f(x)  in  the  form  of  an  infinite  series 
whose  terms  increase  by  powers  of  a;, 

(4)  /(a;)  =A  +  Bx  +  Cx^  +  Bt^  +  Ex^^", 


0.]  ijsrmmm  st^niES  S27 

where  J.,  B,  0,  J),  U^    "^  are  definite  but  as  yet  unknown 
numbers.     As   soon   as   the   existence  of   such  a  series   is 
established,  our  whole  problem  resolves  itself  into  the  deter- 
mination of  the  values  of  J.,  B^  (7,  i>,  U,  .... 
Putting  a;  =  0  in  (4),  then 

/(0)  =  ^, 

that  is,  A  is  the  value  which  the  function  assumes  when 
a;  =  0,  just  as  in  formulae  (2)  and  (3)  the  constant  term  1 
of  the  right  side  gives  the  length  of  the  rod  for  the  tempera- 
ture ^  =  0.  If  we  now  take  the  derivative  of  each  member 
of  equation  (4),  we  have 

(5)  f(x}  =  B-\-2  0x-{-Sl)x^  +  4:Ua^-\- ..., 
and  if  we  now  put  a^  =  0,  we  find 

(6)  fi(i}  =  B;    5  =  -^, 

that  is,  B  is  equal  to  the  value  which  the  derivative  f(x} 
assumes  when  x  is  given  the  value  zero.  On  forming  the 
derivative  of  equation  (5),  we  have 

(7)  f"(x}  =  20-{-2'nDx  +  S'^Ux^i- ..-, 
and,  after  putting  x=0^  we  obtain 

(8)  /'(0)  =  2C;   0=^^, 

where  /"(O)  denotes  the  value  which  f"(po)  assumes  for 
a:  =  0.     If  we  differentiate  again,  we  find 

(9)  f"(x)  =  1.2.3i>  +  2.3.4J5;a;+  -, 
that  is, 

(10)  /"'(0)  =  1.2.3i);   I>  =  -Y^ 


328  CALCULUS  [Ch.  X. 

etc.,  etc.  In  this  way  we  have  determined  the  unknown 
coelhcients  in  a  simple  manner ;  and  for  the  function  in 
question,  f(x)^  we  have  the  series 

(11)  f{x)  =  /(O)  4  I  /'(O)  +  ll  /"(O)  +  I?  /'''(O)  +  ..., 

which  is  called  Maclaurin's  Series.* 

It  is  a  series  of  great  fruitf ulness  and  importance,  yielding 
expansions  by  whose  aid  the  approximate  value  of  many 
functions  can  readily  be  computed  for  any  given  value  of 
the  variable. 

Art.  10.   The  series  for  e*,  sin  a?  and  cos  a?.     We  at  once 

apply  Maclaurin's  Series  to  several  simple  functions. 

I.    Let  f(x)  =  e'^. 

The  successive  derivatives  f  are 

and 

/(O)  =  1,  /'(O)  =  1,  /"(O)  =  1,  /^''(O)  =  1,  .... 

Maclaurin's  Series  accordingly  assumes  the  form 

*  Colin  Maclaurin  (1698-1746)  had  even  at  the  age  of  15  discovered  many 
of  the  theorems  which  he  published  later,  and  before  he  had  attained  the 
age  of  20  was  appointed  Professor  of  Mathematics  at  Aberdeen ;  from 
1725  to  1745  he  was  Professor  of  Mathematics  at  Edinburgh,  and  joined  in 
the  defense  of  the  city  against  the  "Young  Pretender"  in  1745  ;  upon  the 
capture  of  the  city  by  the  latter,  he  fled  to  York,  where  he  died  in  1746. 
The  series  which  we  have  treated  appeared  in  the  Treatise  of  Fluxions^  1742, 
and  is  a  special  case  of  Taylor's  Series,  which  we  shall  take  up  later,  having 
been  recognized  as  such  by  Maclaurin. 

t  In  applying  Maclaurin's  Series,  it  is  better  to  form  all  the  derivatives 
needed  first  (simplifying  as  much  as  possible  at  each  step),  and  then  to  put 
ac  =  0  in  each. 


9-10.]  INFINITE  SERIES  320 

which  is  called  the  exponential  series.  With  its  aid  we  can 
compute  the  value  of  e"^  for  a  given  value  of  x ,  for  x  =  l^ 
we  have  the  series  already  deduced  for  e  (p.  323). 

II.  If  we  put 

(2)  /(^)  =  sin  X, 

we  have  the  derivatives  in  order  as  follows  : 
f\x^    =  cos  x^        /"(.^^  =  —  sin  x^ 
f"(x^  =  —  cos  x^   f^^^C^y  =  sill  ^9   /^(^)  =  cos  X ; 

whence 

/(O)  =/"(0)=/-(0)  =  -  =  0, 

/(O) = 1,  /'"(o)  =  - 1,  rw  =  1  - ; 

we  therefore  have 

/y*  /y»0  /y»5  /y»7  0^*7 

(3)  «ia.  =  ---  +  ---  +  -.... 

III.  Analogously  we  find  for 

(4)  f(x}  =  cos  X, 

/'(a?)  =  —  sin  x^  /'^(^)  =  —  cos  x,  f"'C^^  —  ^^^  ^? 
/iv(2:)  =  cos  a;,       /^(^)  =  —  sin  a; ; 
whence 

/(0)  =  i,  r(0)  =  -i,  /^v(0)  =  i..., 

and  we  have  for  cos  a:  the  following  series  : 

/y*A  /yA  /yfy 

(5)  cosa;=l--  +  ---.... 

It  is  to  be  observed  that,  according  to  the  conventions  of 
p.  74,  X  is  always  to  be  considered  as  the  magnitude  of  the 
angle   in  circular   measure.     If,  for  example,  we   compute 


830  CALCULUS  [Ch.  X. 

the  values  of  sin  1  and  cos  1,  that  is,  the  sine  and  cosine  of 
the  unit  angle  of  circular  measure,  the  radian  (p.  74),  we 
find 

eosl  =  l-2T  +  4T-eT  +  8T-10T  +  -' 

Using  the  numerical  values  calculated  on  p.  138,  we  have 

'l  =  l  ^  =  0.5  1  =  1  i^=:0.1667 

J- =0.0417      i-=  0.0014      -1=0.0083      1=0.0002 
41  6!      5!      7!      


1=0.0000 


0.5014  1.0083  0.1669 


1.0417 

that  is,  we  find 

cos  1  =  1.0417  -  0.5014  =  0.5403, 

sin  1  =  1.0083  -  0.1669  =  0.8414, 

while  the  values  correct  to  four  places  of  decimals  are  0.5403 
and  0.8415,  respectively.  The  fourth  place  of  our  result  is 
thus  correct  in  the  first  case,  and  in  the  second  case  the 
difference  amounts  merely  to  a  unit  in  the  fourth  place.* 

These  are  the  results  of  making  numerical  substitutions  in 
the  series,  but  before  we  have  confidence  in  their  correctness 

*  In  making  computations  of  approximate  values  by  means  of  infinite 
series,  it  is  better  to  carry  the  decimal  approximation  for  each  term  several 
places  of  decimals  further  than  the  number  of  places  to  which  the  result  is 
desired  to  be  correct,  and  to  compute  the  successive  terms  of  the  series  until 
they  no  longer  present  any  significant  figures.  In  the  value  thus  obtained 
the  last  figure  will  always  be  untrustworthy,  but  unless  the  series  is  quite 
slowly  convergent,  all  but  the  last  two  figures  will  be  correct. 


10.]  INFINITE  SERIES  331 

we  must  ascertain  whether  these  values  of  x  (and,  in  general, 
what  values  of  x}  make  our  series  convergent.  According 
to  pp.  318-20,  any  series  whatever  is  certainly  convergent  if, 
from  a  certain  term  on,  the  quotient  of  «,^+i  and  a,^  is  always 
less  than  some  nuniber  which  is  itself  less  than  unity.  In 
the  series  for  e^  the  terms  a^  and  a^+i  have  the  values 


X 

^«  =  — 7    ^nd 

«^r,+l 

nl 

consequently. 

«.+!  _          ^"^^ 

x"" 

a«        (n-\-l)i 

n\ 

in-\-l) 


r 


and  no  matter  what  the  value  of  tjie  number  rr,  this  quotient, 
when  n  increases  without  bound,  is  not  only,  from  a  certain 
term  on,  always  less  than  some  number  which  is  itself  less 
than  unity,  but  indeed  approaches  the  value  zero.  If,  for 
example,  x  =  100,  then  when  n  receives  the  successive  values 
100,  101,  102,  •••,  that  is,  from  the  hundredth  term  on,  the 
quotients  are,  respectively, 

1_Q0       10.0       10.0.       ... 
101'      102'      103'  ' 

and  the  series  is  undoubtedly  convergent.  Naturally,  the 
greater  the  value  of  x,  the  greater  the  number  of  terms 
of  the  series  needed  for  the  numerical  computation  of  an 
approximate  value  of  e^. 

The  series  (3)  and  (5)  may  likewise  be  proved  convergent 
for  all  values  of  x  by  means  of  the  theorems  which  we  have 
established.  We  leave  this  as  an  exercise  for  the  reader. 
We  need  to  employ  the  series  for  sine  and  cosine  only  for 
angles  which  lie  between  0  and  — ;  for  the  values  of  the 
functions  for  all  other  angles  can  be  calculated  from  these.* 

*  Formulse  13  et  seq.,  Appendix. 


332  CALCULUS  [Ch.  X. 

In  this  way  the  values  of  sin  x  and  cos  x  may  actually  be 
determined,  and  we  see  from  the  above  example  that  but  few 
terms  are  needed  to  obtain  fairly  correct  results. 

EXERCISES  XXXVI 
By  means  of  Maclauriii's  Theorem,  show  that 

1.  coiimx  =  l ■ -\ — h  •••. 

2!  4!  6! 

2.  a-  =  1  +xloga  +  ^(\ogay  +  f.  (log ay  +  .••. 

^1  x^      2x* 

3.  logcos:r  =  -  — -  —  -•... 

4.  Va  +  x  =  a'^  +  la^x+  •••. 

5.  sec  a:  =  1  +^  +  5^  +  .... 

2!       4! 

6.  v/e^T^  =  (1  +  a)^ -+ ^+-- 

3(l  +  a)^     9(1  + a)* 

7.  sin^x^a;^-  ^^-  +  _^+.... 

8.  cos^x  =  1  —  x^  -\ 1-  •••. 

Hint  :   Use  Formula  36,  Appendix. 

9.  e«inx  =  l^x+  —  --  +  .... 

2       8 

10.   c^ sm  X  =  X  +  x2  H \-  .... 

3       5!        6!         7!         9! 


Art.  11.  The  series  for  tan  a?.  To  deduce  the  series  for 
tan  X  by  the  aid  of  Maclaurin's  Theorem,  we  need  the  values 
which  tan  x  and  its  higher  derivatives  assume  for  x=  0. 
These  values  cannot  be  so  directly  determined  as  was  the 
case  with  e^^  sin  x,  and  cos  x^  hence  we  make  use  of  an  artifice 


10-11.]  INFINITE  SERIES  333 

which  may  also  be  employed  to  advantage  in  the  develop- 
ment of  other  functions. 

If  we  put 

..-.  .  sin  X 

(1)  V  =  tan  X  = ■'> 

cos  X 

or 

(2)  1/  cos  X  =  sin  x, 

and  using  the  notation  (p.  273), 

we  find,  by  successive  differentiation, 

(3)  y^  cos  a;  —  ^  sin  a^  =  cos  x ; 

(4)  ^''  cos  a;  —  2  y  sin  a^  —  ^  cos  :?;  =  —  sin  a; ; 

(5)  y  cos  X  —  ^  y"  sin  x—  %  y^  cos  x  -\-  y  sin  rr  =  —  cos  a;. 

By  continuing  the  process  of  differentiation,  it  soon  ap- 
pears that,  however  often  we  differentiate,  the  numerical 
coefficients  of  the  left  member  follow  the  same  law  of  forma- 
tion as  those  of  the  Binomial  Series,*  so  that  after  n  differ- 
entiations the  left  member  would  assume  the  form 

(6)  1  •  L 

,  n(n  —  V)(n  —  2)    /«_q)    . 

while  the  right  member  is  the  nth  derivative  of  sin  x 
(p.  274).  The  proof  can  easily  be  made  by  mathematical 
induction.! 

*  Formula  3,  Appendix. 

t  A  similar  relation  holds  for  the  nth  derivative  of  the  product  of  any  two 
functions.  The  general  theorem  is  due  to  Leibnitz,  and  is  usually  called 
Leibnitz's  Theorem. 


334  CALCULUS  [Ch.  X. 

From  these  expressions  we  can  find  the  values  which  tan  x 
and  its  derivatives  assume  for  :r  =  0.     If  they  be  denoted  by 

(y)o,  (y')o,  iy"\,  {y"'\  -, 

it  follows  from  (2),  (3),  (4),  etc.,  that 

(y)o  =  (y')o  =  (y')o=-  =  0, 
(/).  =  !,    (y")o=2,    (^0o=16,    •••; 

(7) 


whence 


tana: 

=  ?  + 

a^ 

2 

1 

1 

.2 

•3 

X 

a^ 

2a:S 

+ 

+ 

~1 

3 

15 

1.2.8-4.5 


.  16  + 


Art.  12.  Taylor's  Theorem.  To  determine  the  series  for 
e^,  sin  a;,  and  cos  x^  we  had  to  make  use  of  the  values 
which  these  functions  and  their  derivatives  assume  when 
a;  =  0.  Every  function,  however,  cannot  be  treated  in  this 
way  ;  for  example,  log  x  and  all  of  its  derivatives  grow 
large  without  bound  when  a;  =  0.  In  order  to  develop  such 
functions  into  series,  we  make  use  of  another  series  due  to 
Taylor,  of  which  Maclaurin's  Series  is  a  special  case. 

Just  as  Maclaurin's  Series  operates  with  the  values  which 

the  function  and  its  derivatives  assume  when  a;  =  0,  Taylor's 

Series  is  based  upon  the  values  which  the  function  and  its 

derivatives   take  when  x  =  h^  h  being  any  given  quantity. 

It  gives  the  value  of  the  function  f(x  +  7i)  when  all  the 

values 

f(K),  fiK),  f"(K),   ..., 

are  known.  Taylor's  Series  can  be  deduced  in  exactly  the 
same  way  as  Maclaurin's.  We  start  with  the  assumption 
that  we  may  express  f(x  -\-  K)  in  the  form 

(1)  f(x  +  K)  =  A  +  Bx  +  Cx'^-\-D:x^  +  E:(^+.>'', 


11-12.]  INFINITE  SERIES  335 

putting  x  +  h  —  y,  we  have 

fQ,)=A  +  Biy-K)  +  Oiy-hy  +  Diy~hy  +  Eiy-hy+.... 

Differentiating  successively  with  respect  to  y,  we  have 

/'(y)    =i?  +  2C(«/-/0  +  3i)(y-A)2  +  4^(y-A)3+..., 
f'iy)  =2C+2-SD(iy-h)  +  S-iE(y-hy+:; 
/"'(y)  =  2.32)  +  2.3.4^(y-A)  +  ..., 

Putting  y  =  h^  i.e.  a:  =  0,  we  obtain 


Substituting  the  values  of  A,  B,  C,  i>,  •••,  thus  obtained 
in  (1),  it  becomes 

(2)  f(ix^  +  h)  =f(h)  +fih)oo  +  £^x'-  +  ^f^oc^  +  -. 

This  is  the  expansion  known  as  Taylor's  Series.*  By 
interchanging  x  and  A,  it  may  also  be  written 

(3)  f(x+h-)^f(x)+fix-)h  +  ^^h^  +  i^^h^+.-. 

The  error  committed  by  discontinuing  Taylor's  or  Mac- 
laurin's  Series  after  any  term  is  equal  to  the  sum  of  all  the 
omitted  terms.  The  value  of  this  sum  or  remainder  can  be 
estimated  or  expressed  in  a  formula.  It  would  be  beyond 
the  scope  of  this  book,  however,  to  take  up  the  determina- 

*  Published  by  Taylor  in  1715  in  his  Metliochis  Incrementoriim.     Brook 
Taylor  (1685-1731),  Doctor  of  Laws,  although  a  jurist,  devoted  considerable 
attention  to  mathematics,  and  was  proficient  in  music  and  painting  as  well, 
23 


336  CALCULUS  [Ch.  X. 

tion  of  the  term  equivalent  to  the  remainder  of  the  series 
after  any  given  term.  The  plan  of  this  book  likewise  does 
not  permit  us  to  take  up  the  question  of  the  convergency  of 
Taylor's  Series,  or  the  proof  that  the  assumption  from  which 
we  started  was  correct.  These  and  other  questions  remain 
unsettled  above,  and  our  discussions  must  be  characterized 
rather  as  making  the  truth  of  the  theorem  plausible  than  as 
a  rigorous  proof.  Strict  determinations  of  all  the  ques- 
tions involved  have,  however,  been  made,  including  the 
values  of  x  for  which  the  series  converges. 

To  show  the  ease  and  rapidity  with  which  a  function  may 
be  developed  into  a  series  by  the  aid  of  Taylor's  Theorem, 
we  carry  out  the  development  of  a  function  first  accord- 
ing to  the  rules  of  algebra  and  then  by  the  theorem  under 
discussion. 

Suppose  we  have  given  the  function 

(4)  y  =f{x)  =  a:r3  +  hx^  -{- ex  +  d. 
If  X  receive  the  increment  A,  we  have 

(5)  f(x  +  h);=  a(x  +  hy  +  h{x  -f-  hy  +  c{x  +  K)-^d', 

the  right-hand  member  of  this  equation  when  expanded 
becomes 

f{x^}i)  =  a7^^^ax^h^-Zax¥-\-ah^ 

+     hx^     ^-Ibxh  -\-hW' 

-\-     ex      -\-  eh 

+  d, 

or  (6)     f(x  +  h)  =  (a3^  -^  hx^  +  ex  -{-  d)  +  {S  ax^  +  2bx  +  c)h 

+  (nax  +  b)h'^  +  ah^ 


12-13.]  INFINITE  SERIES  337 

If  we  apply  Taylor's  Theorem  to  the  function  in  hand, 

viz. : 

f(^x)  =  aa^  -j-  bx^  -{-  ex  -{-  d, 

we  find  /'(^)  =  ^  ^^^  -\-2bx  -{-  c, 

f"(x)  ==^ax  +  2h',  ^^^=S  ax  +  5, 

f'\x}  =  0. 

Substituting  these  values  in  equation  (2),  we  have 

O)  /(^  +  h)  =  (^ax^  +  bx^  -^cx-\-d)-{-(S  ax^  -\-2bx  +  c)h 
+  (3  ao;  +  b)  ¥  +  ah% 

the  same  series  which  was  found  above  by  purely  algebraical 
methods.  The  series  terminates,  because  the  derivatives 
from  the  fourth  on  are  all  zero.  This  example  shows  that 
even  simple  processes  of  algebra  may  be  performed  more 
expeditiously  by  the  use  of  Taylor's  Series,  though,  of 
course,  the  chief  value  of  the  series  is  for  the  expansion 
of  functions  which  are  quite  beyond  the  reach  of  algebra. 


Art.  13.  The  logarithmic 

series.     Let 

(1) 

/(^) 

=  log  X ; 

then  (p.  274) 

(2)      fix)  =1, 

X 

'     /"'(-)  =  i^' 

/"(-)=-i 

n^-)-  ^■\'. 

338  CALCULUS  [Ch.  X. 

We  put  a;  =  1,  since  this  gives  the  foregoing  derivatives 
the  simplest  form,  and  makes  the  computation  of  the  series 
the  most  convenient.     Then 

/(I)  =  0,  /'(I)  =  1,  /"(I)  =  -  1,  /'"(I)  =  1-2, 
/'Xl)  =  -l-2.3,    ..., 

and  on  substituting  these  values  in  Taylor's  formula,  we 
have 

(3)  iog(l  +  /0  =  J-|  +  |-|  +  .... 

This  series  can  be  employed  in  the  determination  of  the 
logarithms  of  numbers  greater  than  unity,  provided  the 
series  is  convergent.  The  logarithms  of  numbers  less  than 
unity  are  obtained  by  giving  h  in  the  last  equation  a  nega- 
tive value,  so  that 

(4)  iog(i-;i)  =  -^-|-|-|-.... 

The  circumstance  that  all  the  signs  are  now  negative 
agrees  with  the  fact  that  the  logarithms  of  numbers  less 
than  unity  have  negative  values. 

According  to  p.  322,  each  of  the  foregoing  series  must 
certainly  converge  as  soon  as  A  <  1,  since  every  term  of  each 
of  the  two  series  is  smaller  than  the  corresponding  term  of 
a  geometric  series  with  the  ratio  h.  If  A  >  1,  the  series 
is  divergent  by  p.  321.  Let  the  student  make  the  proof 
in  detail. 
.    If  we  put  A=l,  equation  (4)  gives  as  the  value  of  log  (0), 

iogO  =  -;i  +  i  +  i  +  i  +  i  +  ...|. 

As  is  well  known,  log  0  is  negative  and  infinitely  large, 
and  we  have  already  convinced  ourselves  (p.  312)  that  the 


13.]  INFINITE  SERIES  339 

series  in  the  right  member  is  divergent.  Equation  (3),  on 
the  other  hand,  yields,  when  ^  =  1,  the  series 

which  is  convergent  (p.  313).  This  series  gives  the  value 
of  log  2,  and  its  sum  is  approximately  0.69325. 

The  series  above  cannot  be  used  at  all  to  compute  log- 
arithms of  numbers  greater  than  2 ;  even  for  numbers  less 
than  2,  it  converges  slowly. 

The  series  which  is  ordinarily  used  in  the  computation 
of  logarithms,  and  which  is  convergent  for  all  values  of 
the  variable  greater  than  unity,  is  deduced  as  follows  : 

Since        log  j-±|  =  log  (1  +  A)  -  log  (1  -  A), 

we  get  by  subtracting  (4)  from  (3), 

This  series  is  convergent  if  A  <  1,  since  it  is  the  difference 
of  two  series,  both  of  which  are  convergent  if  A  <  1.  If, 
now,  iVbe  any  number  greater  than  unity,  we  may  put 

1+h 


whence  h  = 


1-h 

jsr-1 


jsr+1 

which  is  always  a  proper  fraction,  and  will  therefore  make 
the  series  above  convergent. 

Substituting  this  value  of  h  in  the  series  above,  we  should 
obtain  a  series  for  log  iV  convergent  for  all  values  of  iV^ 
greater  than  unity.  The  series  so  obtained  does  not,  how- 
ever, converge  very  rapidly,  and  another  converging  much 
more  rapidly  is  obtained  as  follows  : 


840  CALCULUS  [Ch.  X. 

^4-1       1  4-^ 


Put 


then  h  = 


n         1  —  h 
1 


Here  again  h  is  less  than  unity  for  all  values  of  n  greater 
than  unity,  and  therefore  this  value  of  h  will  make  the 
series  above  convergent.     By  substituting,  we  obtain 

logf!i±IV2|-:^  + 1 +'       ^      '^...l, 


or 

WCwH-l)  =  log^+2| — - —  + + 3 +•••}. 

^^   -^  J        ^     -f    Ivj/i  +  l     8(2n  +  l)3^5(2/i  +  l/^      J 

This  series  is  rapidly  convergent  and  enables  us  to  com- 
pute readily  the  value  of  log  (w  +  1)  when  that  of  log  n  is 
known.     Log  1  being  0,  we  compute  log  2,  then  log  3,  etc. 

These  are  logarithms  to  the  base  g,  since  in  our  differ- 
entiations we  assumed  that  we  were  dealing  with  this  base. 
The  mode  of  passing  from  logarithms  to  the  base  e  to  those 
to  any  other  base,  10,  for  instance,  as  well  as  the  details  of 
the  actual  computations^  are  explained  in  works  on  trigo- 
nometry. / 

Art.  14.  The  Binomial  Theorem.     Given 

(1)  /(^)=^^ 

where  n  may  he  positive  or  negative^  an  integer  or  a  fraction, 
to  find  (x  +  hy. 

By  differentiation 

f  (x)  =  na7""\ 

f"(x)  =  n(n  -  l)(n  -  2)x^-\ 
etc.,  etc. 


13-14.]  INFINITE  SERIES  ,  341 

From  these  values, 

f"(h)  =  n(n  -  l)(n  -  2)/^"-^ 

Applying  Taylor's  Series  to  the  expansion  of  (a:  +  A)",  and 
using  these  values,  we  have 

O  ! 

If  A  =  1,  this  expression  becomes 
(2) 

This  series  is  known  as  the  Binomial  Series.  It  is  true 
for  every  value  of  ?^,  positive  or  negative,  integral  or  frac- 
tional. There  is  a  difference  in  form  between  this  series  and 
the  Binomial  Theorem  for  positive  integral  exponents.  The 
number  of  terms  of  series  (2)  is  in  general  boundless  ;  the 
coefficients  are  in  general  all  different  from  zero  ;  they  can 
assume  the  value  zero  only  when  one  of  the  numbers 

in-\),    (^-2),    (^-3),    (ri-4),    ..., 

can  become  zero ;  that  is,  n  must  be  a  positive  integer.  In 
that  case  the  series  terminates,  and  has  a  finite  number  of 
terms. 

It  can  be  proved  without  much  difficulty  that  whatever 
the  value  of  w,  our  series  converges,  if  x  is  less  than  unity. 
The  proof  is  left  to  the  student  as  an  exercise. 


342  .  CALCULUS  [Ch.  X. 

EXERCISES  XXXVII 

By  means  of  Taylor's  Theorem,  show  that 

1.   sin  (^x  -\-  K)  =  sin  x  -\-  h  cos  x  —  —  sin  a:  —  —  cos  x  +  •••. 
.   2.   log(a  +  a:)=loga+ TlTi  + -TTT +*'*• 
3.   COS  (a  +  x)  =  cos  a  —  X  sin  a  —  —  cos  a  +  —  sm  a  +  •••. 

<^ !  o  ! 

z !          o ! 
5.   log  sin  (a  +  a;)  =  log  sin  a;  +  «cotar  —  — cosec^a:  H cotarcosec^a:  +  •••. 

Art.  15.  Integration  by  series.  Determinations  of  inte- 
grals based  upon  developments  into  series  occur  quite  often 
in  the  applications  of  the  Calculus;  we  must  always  have 
recourse  to  them  when  other  methods  are  not  available. 
This  mode  of  finding  integrals  is  known  as  integration  by 
series. 

The  mode  of  procedure  is  as  follows  :  We  assume  that 
we  are  able  to  develop  the  function  f(x)^  occurring  in  the 
integral 

into  a  convergent  series  ^  arranged  according  to  ascending 
powers  of  x ;  thus,  the  integration  of 

(1)  /(^)  =  «o  +  ^1^  +  ^2^^  +  %^  "• 

gives       I  f(pc)  dx.=  (  (<2q  +  a^x  +  a^x^  +  a.^x^  -\ — ^dx 

=  \  a^dx  +  I  a^xdx  +  1  a^x^dx  +  •••, 
whence 

(2)  J  f(x)  dx  =  a^x  +  a^^  +  a^—  +  ^sj  ^ ^^' 


14-15.]  INFINITE  SERIES  348 

We  note  that  the  resulting  series  (2)  is  undoubtedly 
convergent  for  all  values  of  x  that  make  series  (1)  con- 
vergent.    We  need  merely  to  write  it  in  the  form 

to  see  that  according  to  p.  322  the  expression  in  parenthesis 
is  convergent,  and  hence  series  (2)  also. 

We  pass  at  once  to  some  examples.     Let  the  first  be 


(3)  /jf 


dx 


dx 


According  to  p.  314,  when  a:^  <  1, 

(4)  — ^  =  1  _  ^2  ^  ^4  _  ^6  ^  ^8 ^ 

1  +  x^ 

and 

(5)  J_^^=J(l-:r2+2:4-a;6  +  :^:8...) 

/y»0  /y»0  /y»rf  />^y 

We  proceed  to  draw  an  important  conclusion  from  this 
equation.  According  to  p.  176,  the  integral  of  the  left 
member  is  equal  to  arc  tanrz:;  we  thus  obtain  the  equation 

,  0^   ,   0^       x^    ,   x^        ,     n 

arc  tan x  =  x—  —  ■\-- —  -^^-^ 1"  ^• 

o       5       7       9 

To  determine  the  constant  (7,  we  put  a;  =  0  ;  it  is  sufficient 
to  consider  arc  tan  x  to  be  the  smallest  positive  angle  whose 
tangent  is  x^  for  when  we  know  one  of  the  angles  having 
a  given  tangent,  we  have  trigonometric  formulae  which 
enable  us  readily  to  determine  all  others.     With  this  restric- 


344  CALCULUS  [Ch.  X. 

tion,  we  have  arc  tan  0  =  0,  and  hence   (7=0.     We  have, 
therefore, 

/vO  /v^  /y>7  /vw 

(6)  arc  tan  a;  =  a:  -  —  +  -^  —  Y  4-  —  ;••• 

In  this  way  we  have  developed  arc  tana;  into  a  series, 
called  Gregory's  Series,*  which,  if  obtained  by  Maclaurin's 
Theorem,  would  have  required  complicated  computations, 
since  the  higher  derivatives  of  arc  tan  x  would  then  have 
been  needed. 

If  we  make  a;  =  1,  then  arc  tan  1  is  the  length  of  the  arc 
whose  tangent  is  equal  to  unity ;  i.e.  the  arc  --•  We  obtain, 
accordingly,  from  equation  (4)  for  a;  =  1, 

(5)  4=^-3  +  5-7  +  9"n  +  -' 

and  this  series  having  -  as  its  limit  may  be  used  to  deter- 

4 

mine  values  differing  little  at  will  from  the  true  value  of  —  • 
To  compute  tt,  it  is  best  to  write  the  series  in  the  form 

9  2  2  2 


1-3      5-7      9.  11      13.15 
and 

(6)  -^_1_+    1.1.1 


8      1-3      5-7      9-11      13-15 
As  a  second  example,  we  take  the  integral 

dx 


(T)  f 


VI  -  x'^ 


*  James  Gregory  (1638-1675),  an  English  mathematician,  made  important 
contributions  to  the  development  of  this  theory  of  infinite  series.  The  term 
convergent  was  introduced  by  him. 


15.]  INFINITE  SERIES  345 

By  the  Binomial  Theorem, 

(8) 

1        _  (i_|_l.  ^^_^1     3       2:4    .  1     3     5         a^       ,       ) 


Vr^^      (         212     2     1-22     2     2     1.2.3 
and,  after  integration, 


-^  vr^^^    ^        -^  2    1        -^  2    2    1  . 


-dx  +  '"+C 


1  r^       1  .  3  r^       1  .  3  .  _5  ^7  _ 

^231.251.2.3  7 

According  to  p.  176  the  integral  of  the  left  member  is 
equal  to  arc  sin  x  ;  therefore  Ave  obtain  the  equation 

1  ^3      1.3^      i  .  a  .  5  ^7 
arcsina.  =  a:  +  2|  +  2_||  +  a_|_||  +  ...  +  a 

To  determine  the  constant  we  put  x  =  0  \  and  find  (under 
the  same  restriction  as  was  imposed  on  arc  tan  a;),  that  (7=0 
also,  and  hence  l^ve 

'         1 3;3        1  •.  3       ^5        1.3.5^      ^7 

(9)      arcsin:r  =  :.  +  ||  +  |-|.|4-^-|-|.|+-. 
If  we  put  ^  =  J,  then  arc  sin  ^  =  ^.     We  therefore  have 

rjr.  1  1     .    /1A3  1    .     3_    .    (l\b  .1     .     a    .    5.    (X\I 

riO^  —        I    2     \l)     I    2 2 L2J_-l-2 2_L2_L^2__i_  ... 

^     ^     6      21.31.2.51.2.3.7  * 

This  series  is  better  adapted  to  the  computation  of  tt  than 
the  previous  one. 

The  computation  is  made  still  more  readily  by  means  of 
Gregory's  Series   applied   to  one   of   certain   trigonometric 


M6  CALCULUS  [Ch.  X. 

relations,  of  which  we  give  the  following,  due  to  Gauss,*  as 
a  specimen  : 

(11)     —  =  12  arc  tan  ^^  +  8  arc  tan  ^j  —  5  arc  tan  ^^-^, 

The  correctness  of  this  relation  may  be  verified  by  the 
methods  of  trigonometry,  and  it  will  be  necessary  to  use  but 
a  few  terms  of  the  series  for  arc  tan  x  to  obtain  quite  a  close 
approximation  to  the  value  of  ir. 

We  also  see  how  much  superior  such  methods  as  these 
for  the  calculation  of  ir  are  to  those  applied  in  elementary 
geometry. 

Exercise.  By  means  of  (11)  calculate  the  value  of  ir  to  8  or  more 
decimal  places.  Ans.  tt  =  8.141,  592,  653,  589,  793,  238,  462,  643,  383, 
279,  502,  884,  ....     .     , 

Art.  16.  Table  of  series.  We  collect  into  a  table  the 
principal  series  which  we  have  considered  : 

1.  f{x)  =  /(O)  +  xf'(0)  + 1^  /"(O)  + 1^  f"'(0)  +  .... 

^  *  '  (Maclaurin's  Series.) 

2.  fix  +  h)  =  fix)  +  hf'ix)  + 1^  f"(x)  + 1^  f"'(x)  +  .... 

(Taylor's  Series.) 

3.  (1  +  ^)"  ^  1  +  nx  +  ^^^  -  ^)  ^^  +  ^^^^  -  W^""  --^  x^  +  ....    ^ 

(The  Binomial  Series.) 


vr^^      2     2.4     2-4.6 


*  Carl  Friedrich  Gauss  (1777-1855),  the  greatest  of  mathematicians, 
princeps  mathematicorum,  Professor  in  the  University  of  Gottingen,  enriched 
all  branches  of  mathematics,  both  pure  and  applied,  with  the  lasting  fruits  of 
his  wonderful  genius.  His  classic  work  on  the  Theory  of  Numbers,  Disqni- 
sitiones  Arithmeticae^  was  published  (1801)  when  he  was  only  twenty-four 
years  of  age.     With  Weber,  he  invented  the  first  electric  telegraph. 


16-17.]  INFINITE  SERIES  347 

/V»^  />»*  /^n^ 

6.cos.  =  l-f^  +  |^_|.  +  .... 

7.  arc  sin  a?  =  a?  +  - — ra^^  +  ^r — ^-^a?^  +  '". 

z  *  6  2  •  4  •  5 

8.  arctanic=a?-^  +  ^-~^  +  «-. 


9.   e*  =  l  +  ic  +  ^+^  +  ^  + 


(Gregory's  Series.) 
2!     3!     4!^'"" 


(Tlie  Exponential  Series.) 

2  3 

10.  a''  =  l  +  3cloga  +  f-\og^a  +  f-\og^a  +  —. 


/y.2        ^^3        ^^4 

11.  log(l  +  a?)=:a?-^  +  |--^  +  ..v 


(The  Logarithmic  Series.) 


Art.  17.   Indeterminate  forms.     We  have  shown  (Foot- 
note, pp.  121-123)  that  the  quotient 


sma; 


approaches  unity  as  x  approaches  zero ;  this  conclusion  can 
be  deduced  directly  by  means  of  our  developments  into 
series.     We  have  (p.  329) 

^  _  ^  _i_  ^"^  _  . . . 
^.  sin  a^      1      3!      5! 


X 


(2)  =1-^!+^!-' 

and  when  x  approaches  zero  the  right  member  approaches 
the  value  1 ;   i.e. 

(3)  ;ro[^]=i- 


348  CALCULUS  [Ch.  X. 

We  may  not  directly  put  a^  =  0  in  the  right  member  of 
(2),  since  to  obtain  (2)  we  divide  by  x. 

This  method  may  be  used  to  find  the  limit  of  other  frac- 
tions for  certain  values  of  the  variable,  for  which  the 
numerator  and  denominator  both  are  equal  to  zero.  Such  a 
fraction  is 

log(l+a;)  ' 

when  a:  =  0,  both  numerator  and  denominator  vanish.  By 
the  developments  into  series  made  on  pp.  341  and  338,  we 
find 

i  (  i      .     ^  .     ^C^  —  1)2.  ) 

^-(^  +  T'^+     1T2    ^+-f 


(4) 


l-(l+:ry 

log  {1  +  x)  X  _x^      a^ 

1  2"  3" 
n  n(n  —  1)  _ 
T         1.2     "" 


2      3 

If  we  now  let  x  approach  zero,  we  see  that  the  fraction  on 
the  right  approaches  —  n ;  that  is, 

We  pass  now  to  the  consideration  of  the  fraction 

(6)  /(^, 

and  deduce  a  general  method  applicable  to  all  such  cases. 

For  convenience,  we  call  the  value  of  x^  which  causes 
a  fraction  to  assume  the  indeterminate  form  -,  a  critical 
value  of  X. 


17.]  INFINITE  SERIES  349 

For  all  values  of  x^  except  the  critical  values,  the  fraction 
has  a  definite  value  which  can  be  determined  by  direct 
substitution  of  the  value  of  x ;  for  the  critical  values  of  a;, 
this  is,  however,  not  the  case,  and,  as  in  the  instances  above, 
so,  in  general,  we  seek  to  determine  the  limit  which  the  frac- 
tion approaches  as  x  approaches  the  critical  value. 

We  assume  that  both  functions  vanish  when  a:  =  a,  so  that 

(7)  /«=0,     (/)(a)=0; 

and,  accordingly,  x=^  a  is  a  critical  value. 
By  Taylor's  Theorem, 

/(a  +  A)  =  /(a)  + Ar(a) +:p^/"(«)+ -, 
and,  inasmuch  as  /(a)  =  0  and  <^(a)  =  0, 


(9)  = 1-2 


<^'(«)+l^<^"(«)  + 


This  expression  is  true  for  all  values  of  h  (except  A  =  0) 
for  which  Taylor's  Series  converges.     If  we  let  A  =  0,  then 

no^  lim  \fix)l._fia) 

^      ^  ^  =  ''U(:»^)J~<^'(«)' 

We  see  thus  that  we  can  find  the  required  limit  of  the 
fractio'h  hy  simply  substituting  for  the  numerator  and  denom- 
inator their  derivatives  with  respect  to  x. 


so  that 


350  CALCULUS  [Ch.  X. 

Art.  18.  Illustrative  examples  of  the  determination  of  the 
limits  of  indeterminate  forms. 

I.  To  find  the  limit  of 

^  ^  ,  a^  +  2x^-x-2 

when  x=l. 

By  substituting  1  for  x  we  see  that  the  fraction  assumes 

the  form  -.     We  have 

f\x')  =  Sx^-12x-\-ll, 

<^'(a;)  =  3^:2 +  4^-1, 
whence  /'(1)=2   and   </>'(!)  =  6, 

/^a)^2_i. 

</)'(l)      6      3' 

»•«•  (12)  .iiL^  +  2^-:._2  J=«- 

II.  To  find  the  limiting  value  of 

(13)  /W^l^l^, 

<^(a:)       logo; 

when  a;  =  1.     We  have 

X 

consequently, 

f(x^      x^  —  1 

III.  For  the  fraction  •;  )  (  = ^ 

<^(^)       logo; 

we  find  /^(:r)=2:z;,    f(2^)  =  i, 

-,  lim   r^2 

and  ^^l' 


r:r2  -  1' 


2^  =  2. 


18.] 


INFINITE  SERIES 


351 


The  development  of  functions  into  series  is  of  great  help 
also  in  the  determination  of  the  limits  of  fractions  in  which 
the  numerator  and  the  denominator  are  infinite,  as  well  as 
of  products,  one  of  whose  factors  is  zero,  and  the  other  is 
infinite  for  the  critical  value. 


IV.    To  determine  (p.  241) 


(15) 

When  a  =  b^ 


lim 
a  =  b 


1 


log 


(a  —  x)h' 


a  —  h         (J>  —  ^)^. 
becomes  infinite,  and 


(16)  log  ^ — ^  =  log  1  =  0. 


We  put 


(h  —  x^a 


so  that 


a  —  X  =  all )    and    b  —  x  =  bll  —-ji 


1    («- 

1^^(6- 

-x}b 
-x)a 

abh- 
ab(l- 

-               1- 

X 

a 

X 

"b 

= 

M'-t) 

-log(l- 

ly  or  (p.  338) 

log^^ 

-x^b 
—  x)a 

-\i 

-ifys--Mt 

2&2^3  63 

■•! 

-<i 

aj       2  W 

ay     8  w 

-s- 

a  — 

h      x^a^-lP' 

1  3?  a^  —  h^ 

.  -I-  ... 

"^    al 

>         2     a2j2 

3     a%^ 

= 

=  ia- 

^l     ab^2 

a  +  b     3?  a^ 
a^h"^       3 

^ab^y^ 

a%^ 

•!■ 

*  Formula  6,  Appendix. 


24 


352  CALCULUS  [Ch.  X. 

Substituting  this  value  in  (15),  we  have 

1      1       (g  —  x)h  _x     J^  ,  ^^  a  +  h      ^  a^H-  ah  +  ^^  , 
a-h   ''^(h-x)a'~l'ab      ^IfiW-       J        '^^  ' 

and  hence 

J^"^r_JL,  w(^-^)  ^V,^.^' .£!.... 

«-^La-6      ^(6-:r)aJ      a2^a3^a4 

=  -.(1+-  +  -^  +  -)^ 
a^\        a      a^  J 

and  by  applying  the  formula  for  the  sum  of  a  geometric 
series  (p.  314), 

n  7)     ^"^  f-^  loff  (^^1^1  =  ^  ^_  = ? 

^      ^     a  =  h\_a-h    ^  {h-x^a]      a^^_x      aQa-x) 

a 
V.    If,  in  the  expression 

1  1 


u  = 


X      log  (1  +  x^ 


x  approach  the  value  zero,  then  both  minuend  and  subtra- 
hend become  infinite,  and  the  expression  is  indeterminate. 
In  order  to  find  its  limiting  value,  we  first  reduce  to  a 
common  denominator, 

_  log  (\-\-  x^  —  x 
X  log  (\-\-  x) 

This  assumes  the  form  -  when  a;  =  0.     The  limit  of  this 

fraction  could  be  found  by  the  method  which  we  have  estab- 

0 
lished  for  the  form  -•     Leaving  this  to  the  student  as  an 

exercise,  we  determine  the  limit  by  using  the  expansion  of 


18.]  INFINITE  SERIES  353 

log  (1  4- a;)    into   a  series.      When    this   is   substituted,    we 

obtain 

X  _x'^      x^      _  x^      a^ 


X        X" 

"'.1-2 


x_^     V 
1       2"V  . 


and  dividing  numerator  and  denominator  by  x\  and  then 
letting  a;  =  0,  we  find  the  required  limiting  value 


(18)     •   jro 


1 1 ^i=-i. 

X      log  (1  +  X)]  2 


VI.    To  find 

lim 

X 


When  x  grows  large  without  bound,  the  numerator  does 
so  likewise.     But  by  the  series  for  e^,  we  have 

£!=i+i+^  +  |^  +  |^  +  ..., 

XX  2 !      3 !      4 ! 

and  this  series  is  infinite  when  x  is  infinite ;  that  is, 

(19)  .-[f]= 


QO. 


If  we  now  put  e^  =  y^  i.e.  a;  =  log  «/,  then   «/ =  oo,  when 

a;  =  00,  and 

and  hence 

When,  then,  y  grows  larger  and  larger,  the  quotient  of 
log  yhjy  approaches  more  and  more  nearly  to  zero. 


354  CALCULUS  [Ch.  X. 

If,  in  the  last  equation,  we  put 

1 

X 

as  ?/  =  Qo,  rr  =  0,  and  we  have 

logv         ^      1 

=  X  loff-  =  —  X  losr  X. 

y  ^x  °    ' 

/ro[-iog.]=;r.[^]=o; 

here  we  have  a  formula  for  the  limiting  value  of  the  product 
X  log  x^  one  of  whose  factors,  x^  approaches  zero,  while  the 
other,  log  ir,  grows  large  without  bound. 

VII.  We  apply  this  result  in  determining 

which  for  the  critical  value  assumes  the  form  0^. 
We  have  * 

Accordingly, 

VIII.  Consider,  next. 


lini 
x  =  () 


This  is  of  the  form  oo^  if  a;  =  0.     We  transform  it  by 
noticing  that  i  =  6"^^^^  hence  we  seek  to  find  J^^^  ^-logx^mx^ 

X 

*  Formula  7,  Appendix. 


18-id.]  i]srPiNtT:BJ  SERIES  355 

T~»    j_     lira    1  •  lira       Sin  x , 

But  ^'^ol^g^«l^^=:t-0^-^lo^^ 

lim       T  /  .  lim    sina;      ^\ 

=  0,  as  shown  above. 
Accordingly, 


lim 
x  =  0 


sin  a: 


IX.    Let  us  take  up  next 

lim    x^^' 
X  ~1 

11  X  =  1,  this  is  of  the  form  1*. 

1 
Let  2/  =  x^-^' 

Then  log  y  =  --^ —      l^^or  x=  l^  this  is  of  the  form   — 
Accordingly, 

.'riD«s»i=,'L">[S]— 1- 

Art.  19.   Types  of  indeterminate  forms.     The  principal 
indeterminate  forms  are 

-;     — ;     0-Qo;     oo  — oo;     0^;     oo^;     1*; 
0       00 

all  of  these  have  been  exemplified  above. 

The  form  occurring  most  often  and  treated  most  simply  is 
the  first.      Generally  the  second   and  third  forms  may  be 


856  CALCULUS  [Ch.  X. 

reduced  to  the  first  form  by  a  simple  transformation.     Any 

A  B 

fraction    -^   may   be  written  in  the  form   -zr-    If  A  and  B 

1       1         ^ 

increase  without  bound,   —  and  —   approach  zero,  so  that 

A  B 

the  indeterminate  form  becomes  —     Also,  if  in  a  product 

AB^  one  of  the  factors,  A^  approaches  zero,  while  the  other,  B^ 
increases  without  bound,  we  may  write  the  product  in  the 

A  0 

form  — ?  which  is  in  the  first  form,  viz.^  — 

^  0 

Usually  functions  in  the  form    -    can  be  evaluated  by 

differentiating  the  numerator  and  the  denominator,  and  sub- 
stituting the  critical  value  in  the  quotient  of  the  results. 
Sometimes,  however,  this  will  not  succeed,  as  the  function 

retains  the  form  -,  no  matter  how  often  the  differentiation 

—1 
is  repeated.      The  function  — ,  as   x  =  co^  is  an  instance. 

e  ^ 

In  such  cases  the  evaluation  may  often  be  accomplished  by 
expansion  into  series. 

It  may  be  proved  without  much  difficulty  that  the  form 
—   can,  like    -,  be    evaluated   by  substituting   the   critical 

value  in  the  quotient  of  the  derivative  of  the  numerator 
by  that  of  the  denominator,  but  we  let  this  simple  mention 
of  this  theorem  suffice. 

EXERCISES  XXXVIII 

Find  the  limits  of  the  following  expressions  : 

1.  ,  as  X  =  5.  3.  ~~  "     " 


x^- 


2.  ,  as  X  =  a.  4.  — ,  as  ar  =  —  |. 

x-a  10a:2-f  29a:  +  10 


19-20.]  INFINITE  SERIES  357 

T-  X*  -{-  X"^  -  4:  X  —  4: 


5. 

*^^^as:.-0. 

X 

6. 

\^^^,  as  X  =  1. 

1   —  X 

7. 

2;5_5^2_^4 

x^  +  2x^+3x-Q 

8. 

-^  as  X  =  0. 
a^  —  1 

9. 

*^"^asx-0. 

Sin  X 

x  =  l. 


,/^    1  —  COS  a:  .  ^ 

10.  ,  as  X  =  0. 

X     ■ 


x^  +  2x^~Sx^-Sx- 

-4 

i.  as  X  =  —  2. 

ii.  as  x  :k-{-2. 

iii.  as  x  =  —  1. 

16. 

l^S^,  asx^l. 

X  —    1 

17. 

— ,  as  X  =1  CO. 
x^ 

18. 

«'-*•,  as  x  =  0. 

1 

11. 


1  —  COS  X  .  Ci  19-   (cos  xY,  as  x  =  0. 
,  as  a:  =  0.  v         /  ' 


12.  ij^^^,  as  x  -  0. 

y/x 


20.  xe'',  as  a:  =  0. 


gx  gSin  a 


21.  ^ ^ as  a:  =  0. 

13.  X  cot  a:,  as  ar  =  0.  x  -  sin  x 

14.  -^,  asx  =  0.  22.   sin(x  +  l)  •  5^^,  as  x=-l. 
cot  X  X  +  1 

Art.  20.  Calculation  with  small  quantities.  An  impor- 
tant practical  application  of  the  expansion  of  functions  into 
series,  given  in  this  chapter,  occurs  in  calculation  with  small 
quantities;  in  such  cases  it  is  generally  sufficient  to  take 
only  the  first  few  terms  of  the  series,  so  that  a  simple  and 
easily  handled  expression  is  obtained  from  the  originally 
infinite  series. 

But  we  must  have  a  'clear  conception  of  what  is  meant  by 
small  quantities.  "Absolutely  small"  quantities  are  non- 
existent as  well  for  the  investigator  in  physical  science  as 
for  the  mathematician.  If  we  endeavor  to  determine  the 
true  capacity  of  a  liter  flask  by  weighing  it  when  full  of 
water  and  when  empty,  a  determination  to  a  ten  thousandth, 
i.e.  weighing  accurately  within  J^  gram,  is  in  most  cases 


358  CALCULUS  [Ch.  X. 

sufficient,  and  we  may  regard  the  last  weight  as  a  small 
quantity.  In  careful  chemical  analyses,  however,  where 
tenths  of  a  milligram  are  of  the  greatest  importance,  an 
error  of  J^  gram  would  make  the  analysis  worthless.  The 
astronomer  in  measuring  the  distances  of  planets  can  neglect 
lengths  of  many  kilometers  as  in  nowise  affecting  the 
accuracy  of  his  results,  while  the  physicist  in  measuring 
the  lengths  of  light  waves  finds  millionths  of  millimeter 
of  decisive  importance  in  observations  and  calculation. 
"  Small  Quantity "  is,  therefore,  a  relative  conception,  and 
we  have  the  right  to  call  a  quantity  small  only  in  comparison 
with  a  second  much  larger  one.  We  may  never  neglect  a 
quantity  in  our  calculations  because  it  appears  to  be  small 
in  itself  (being  only  a  millionth,  say),  but  can  do  so  only 
when  it  occurs  in  connection  with  a  quantity  so  much  larger 
that  the  small  quantity  would  exert  no  influence  upon  the 
degree  of  accuracy  which  we  wish  to  attain.  In  order  to 
express  any  given  quantity  as  the  sum  of  one  or  two  large 
quantities,  and  of  such  negligible  small  quantities,  the  devel- 
opments into  series  often  give  us  valuable  assistance,  as  we 
shall  show  in  some  examples. 

Art.  21.  Reduction  of  barometric  readings  to  0°  C.  The 
length  I  of  a  column  of  mercury  sustained  by  the  atmos- 
phere, the  cross-section  being  constant,  varies  with  the 
temperature  according  to  the  formula 

Z  =  Zq(1  + 0.00018  0, 

where  l^  denotes  the  length  at  the  temperature  ^=0°.     The 

barometric  height  l^  corresponding  to  the  height  I  observed 

at  the   temperature  t  would,  therefore,  be  determined  by 

the  formula,  , 

I  = ? 

^      1  +  0.00018^ 


20-22.]  INFINITE  SERIES  359 

But  according  to  p.  314, 

1  +  a 
or,  if  a  =  0M01St, 

l^  =.  l(\  -  0.00018  t  +  [0.00018  ^2...). 

Now  even  if  ^  =  30°,  the  third  term  of  the  series 
[0.00018  ^]2  is  smaller  than  0.00003,  which  may  be  neglected 
in  comparison  with  unity,  so  that  we  may  write  as  a  quite 
sufficient  approximation, 

?^  =  ?(1- 0.00018  0- 
It  is  generally  advantageous  to  transform  the  equations 
so  as  to  make  the  small  quantities  appear  as  terms  of  a 
sum  in  connection  with  unity.  If  a  calculation  or  obser- 
vation is  to  be  carried  out  accurately  within  one-tenth  of 
one  per  cent,  terms  whose  aggregate  is  less  than  0.001  may 
be  neglected ;  if  an  error  of  two  or  three  per  cent  is  admis- 
sible, terms  less  than  0.01  may  be  neglected,  etc.,  etc. 

Art.  22.  Simplified  hypsometric  formula.  We  found 
(p.  223)  that  the  elevation  H  above  the  earth's  surface  was 

(1)  ^=flog|; 

even  at  elevations  of  1000  meters,  B  is  but  slightly  greater 
than  ^,  so  that  B  —  y  may  be  regarded  as  small  in  compari- 
son with  B  as  well  as  with  y.  If  we  put  equation  (1)  in 
the  form 

and  expand  according  to  p.  338,  we  have 

S        y     \  2y   ) 


360  CALCULUS  [Ch.  X. 

the  terms  involving  higher  powers   of  the   small  quantity 
^  being  neglected ;  in  many  cases  we  may  even  neglect 

if 

the  second  term  within  the  parenthesis. 

We  can  also  write  equation  (1)  in  the  form 

-=-!'- ('-^ 

and  obtain  on  developing  into  series, 

^  ^  S        B     \^      2B    ) 

This  formula  also  can  be  employed  for  moderate  eleva- 
tions ;  but  we  can  get  a  much  better  approximation  by  the 
aid  of  the  following  artifice.  In  formula  (2)  the  corrective 
term  is  negative,  and  in  formula  (3)  it  is  positive,  while  in 

either  case  it  is  about  the  same  ( — ^^  is  but  little  different 

B-v\  ^  ^y 

from         ^  ) ;  the  true  value  lies  therefore  about  midway 
'z  B  ) 

between 

The  two  expressions  differ  only  in  the  denominator,  and  if 
we  introduce  the  average  denominator  ^-^ — ,  we  have 

(4)  ^=2|-fe^' 

S     B  +  y 

which  is  the  formula  most  used  in  practice. 


CHAPTER   XI 
MAXIMA   AND    MINIMA 

Art.  2.   Conditions  for   a   maximum   or  minimum.     The 

accompanying  curve   (Fig.   60),  which   corresponds  to    the 

equation 

(1)  y  =  sin  X, 


r- 
t 

~ 

— 

■"" 

Y 

!Y 

/ 

1 

,1, 

/' 

" 

?~ 

V 

D 

/ 

^ 

ps 

p 

r 

/ 

^ 

V 

s, 

s 

/ 

/ 

\ 

/ 

/ 

0 

I 

lo' 

A 

3 

\z 

A 

TT 

571 

37r 

\ 

^ 

ITT 

y 

Ktt 

Y 

y 

^ 

X 

^ 

3 

^ 

/ 

2 

2 

^ 

r- 

j 

1 

_ 

— 

_ 

1 

1 

Fig.  60. 

reaches  its  highest  position  in  the  points  whose  abscissae 
have  the  values 

TT         5  TT         9  TT 

and  its  lowest  position  where  the  values  of  x  are 

TT         3  TT         7  TT 

"T   ^'   T'"  •••' 

the  first-named  positions  are  called  maxima,  and  the  second, 
minima,  of  the  curve.  The  function  sin  a:  has  therefore 
maximum  values  when  a;  is  equal  to 

-•>       —  H:  Z  TT,        -  ±  4  TT,        •••, 


2      2 


361 


S62  CALCVLV8  ICk.  XI. 

and  minimum  values  when  x  is  equal  to 

At  all  of  the  points  having  these  abscissae  the  tangent  to 
the  curve  is  parallel  to  the  a;-axis;  therefore,  for  all  these 
values  of  x  the  coefficient  m  (p.  25)  of  the  tangent  line  is 
zero ;  that  is,  cos  x  must  be  zero  for  these  values  of  a;,  which 
in  fact  is  the  case. 

These  considerations  may  be  extended  to  any  curve  corre- 
sponding to  an  equation  of  the  form 

(2)  y=f(x). 

We  define  maxima  and  minima  formally  as  follows : 

If  h  denote  a  fixed  number  as  small  as  may  he  necessary^ 
and  if  as  x  increases  from  a  —  h  to  a^  y  =f(x)  also  increases^ 
and  as  x  increases  from  a  to  a  -\-h^  y  decreases^  then  x  =  a  is 
the  abscissa  of  a  7naxhnuin  point  of  y.  Likewise^  if  as  x 
increases  from  a  —  h  to  a^  y  decreases^  and  as  x  increases  from 
atoa-\-h^y  increases^  then  x  =  a  is  the  abscissa  of  a  tninimum 
point  of  y. 

At  every  position  where  a  curve  has  a  maximum  or  mini- 
mum the  tangent  to  the  curve  is  parallel  *  to  the  axis  of 

*  This  is  true  only  when  the  function  and  its  first  derivative  are  con- 
tinuous. If  the  first  derivative  becomes  infinite,  the  tangent  is  perpendicular 
to  the  ic-axis,  and  there  may  be  a  maximum  or  minimum.  In  case  a  maxi- 
mum or  minimum  exists,  the  point  has  other  and  more  characteristic 
properties  than  those  of  maxima  and  minima,  and  it  is  accordingly  usually 
not  classified  with  maxima  and  minima.  There  may  also  be  a  maximum  or 
minimum  if  the  first  derivative  is  discontinuous  without  becoming  infinite. 
By  turning  Fig.  45,  p.  164,  about  0  as  a  pivot,  •■hrough  a  negative  angle 
whose  magnitude  is  greater  than  a  and  less  than  a',  the  point  P  becomes  a 
minimum.  Since  we  have  restricted  ourselves  to  the  consideration  of  con- 
tinuous functions  only,  the  closer  examination  of  these  points  does  not  fall 
within  the  scope  of  our  work. 


1.] 


MAXIMA   AND  MINIMA 


363 


abscissse,  and  hence  the  slope  of  the  tangent  is  equal  to  zero ; 
but  it  must  be  noted  that  the  converse  is  not  always  true. 
Accordingly,  for  all  these  values  of  x^ 


(3) 


di/  _  df{x) 


dx        dx 


f{x-)=0. 


This  is  the  equation  from  which  the  values  of  x  are  calcu- 
lated, for  which  f(x)  may  have  a  maximum  or  minimum. 
As  an  example,  the  derivative  of  the  function 

y  =  2x^-^x^  +  12x-X 

which  is  represented  by  the  accompanying  curve 
(Fig.  61),  is 

•        ^  =  6  2:2  -  18  a;  +  12. 
dxj 

Equating  this  to  zero,  we  have 

6(2^2  -  3  2;  + 2)=  0. 

The    roots    of    this    quadratic    equation    are 
x^  =  l  and  0^2  =  2 ;    the  curve  shows   that  for 
the  first  value  y  presents  a  maximum,  for   the   second   a 
minimum. 

Inasmuch  as  usually  only  the  functions  themselves,  and 
not  the  curves  representing  them,  are  known,  it  is  further 

necessary  to  ascertain  whether  the 
various  values  of  x  which  satisfy 
equation  (3)  actually  correspond  to 
a  maximum  or  minimum  of  the 
function  or  not.  Inspection  of 
Fig.    62   shows   that   the   curve   is 

X      concave  toward  the  a;-axis   at  the 

Fig.  62.  maximum  and  convex  at  the  mini- 


FiG.  61. 


364  CALCULUS  [Ch.  XL 

mum.  Hence,  as  has  been  shown  previously  (p.  278)  for 
any  value  of  x  at  which  a  maximum  occurs, 

g<Oor-g/(.)<0, 

and  for  such  at  which  a  minimum  occurs, 

dx^  ax^ 

In  the  figure,  the  maximum  and  the  minimum  were  both 
above  the  a;-axis,  but  it  may  be  seen  similarly  that  these 
conditions  hold  also  for  maxima  and  minima  which  lie  below 
the  a;-axis. 

The  abscissa,  x,  of  a  maximum  or  a  minimum  must  there- 
fore satisfy  one  of  the  following  sets  of  conditions : 

At  a  minimum,     — —  /(a?)  =  0 ;    — -  f(cc) ,  positive. 
doc  ddc^ 

d  d^ 

At  a  maximum,    — ^  /(a?)  =  0 :    ^— -  /(a?) ,  negative. 
due  anc^ 

It  may  happen  that  both  the  first  and  the  second  deriva- 
tive assume  the  value  zero  for  the  same  value  of  x.  This 
case  will  be  discussed  later  (p.  367). 

In  the  example  above  we  find 

y'=|g=6(2a;-3) 

and  for  x  =  1,  ?/"  =  —  6,  while  for  a;  =  2,  y"  z=  Q.  Hence, 
X  =  1  corresponds  to  a  maximum,  and  rz;  =  2  to  a  minimum, 
just  as  the  figure  indicates. 

Art.  2.  Points  of  inflexion  of  curves.  We  haye  already 
defined  (p.  277)  a  point  of  inflexion  as  a  point  separating 
a  concave  from  a  convex  portion  of  a  curve.     At  such   a 


1-2.] 


MAXIMA   AND  MINIMA 


365 


point  the  curve  is  crossed  by  the  tangent  so  that  it  lies 
partly  on  the  one  side,  partly  on  the  other  of  the  tangent. 
The  tangent  at  a  point  of  inflexion  is  often  called  an 
inflexional  tangent. 

To  fix  oar  ideas,  we  assume  that  the  curve  is  concave 
toward  the  a;-axis  before  the  point  of  inflexion,  and  convex 
toward  the  a;-axis  afterward;  as,  for  instance,  the  curve  in 
P'ig.  63  ;  i.e.  from  A  to  C  the  slope  of  the  tangent  decreases 
continually  (the  angle  t  itself  first 
diminishes  along  AB  through  posi- 
tive acute  angles  to  zero,  then  along 
BC  it  diminishes  from  180°  to  the 
angle  CFX,  the  slope  of  the  inflex- 
ional tangent);  along  the  convex 
portion  from  C  to  ^,  tan  t  contin- 
ually increases,  the  angle  t  increas- 
ing from  OFX  to  180°,  and  then 
from  zero  through  acute  angles  the 
tangent  accordingly  assumes  a  mini- 
mum value  at  C.  Just  the  opposite  would  be  true  if  a  con- 
cave portion  of  the  curve  were  joined  on  at  F;  E  would 
likewise  be  a  point  of  inflexion,  but  at  this  point  the  slope 
of  the  tangent  line  would  assume  a  maximum  value.  It 
is  easily  seen  that  either  one  or  the  other  of  these  cases 
must  always  occur  when  a  curve  has  a  point  of  inflexion. 

To  find  the  points  which  may  be  points  of  inflexion  of  a 
curve,  we  have  to  determine  such  values  of  x  as  will  make 
the  slope  of  the  tangent  (which  is  equal  to  the  first  deriva- 
tive) a  maximum  or  minimum ;  that  is,  the  values  for  which 


Y 

i 

V 

\  ° 

/ 

0 

f\ 

X 

Fig.  63. 


d  tan  y  _  dj/_  _  ^y  _  ^ 
dx  dx      dx^ 


366  CALCULUS  [Ch.  XT. 

In  words,  those  values  of  x  which  cause  the  second  deriva- 
tive to  vanish  (and  no  others^  may  be  abscissa}  of  points  of 
inflexion. 

To  determine  whether  there  actually  is  a  point  of  inflexion 
for  these  values  of  x,  we  must  examine  the  third  derivative 
(i.e.  the  second  derivative  of  the  slope  for  which  we  are 
seeking  the  maxima  and  minima).  If  the  third  derivative 
does  not  become  zero  for  the  value  of  x  in  hand,  this  value 
corresponds  to  a  point  of  inflexion. 

For  the  equation  of  the  curve  considered  above, 


we  find 


^==6(x^-^x-\-2);    ^=6(2a;-3). 
dx  dx^ 


The  abscissa  of .  the  only  point  which  can  be  a  point  of 
inflexion  is  found  by  solving  the  equation 

2  a;  -  3  =  0, 
which  gives  x=  ^,  to  which  y  =  -y-  corresponds  as  ordinate. 
These  are  the  values  of  the  coordinates  of  the  point  of  in- 
flexion (Fig.  61). 

It  may  also  be  noticed  that  if  x  increases  continuously, 
-^  changes  sign  from  positive  to  negative  as  y  passes 
through  a  maximum  value,  and  from  negative  to  positive 

as  y  passes  through'  a  minimum  value;  but  if  -^  does  not 

dx 

change  sign,  y  has  neither  a  maximum  nor  minimum. 

We  have  already  mentioned  (Footnote,  p.  362)  that  some- 
times y  has  a  maximum  or  minimum  value  when  — ^  changes 

dx 

sign  by  passing  through  infinity ;  such  cases,  however, 
demand  special  investigation,  which  the  scope  of  our  work 
does  not  permit  us  to  make. 


2-3.]  MAXIMA   AND  MINIMA  367 

EXERCISES    XXXIX  , 

Examine  the  following  curves  for  points  of  inflexion : 

^-   y=^ ^*  Ans.  X  =  0,  ±aV3. 

a^  +  x^ 

(The  abscissa  only  of  the  point  is  given  in  each  answer.) 

2.  y  =  '^-  Ans.  x  =  0.         ^'   y^  =  '^P^'  ^^n...  None. 

o  .3    o        Q  A  1  6.    ?/ =  tan  x.        Ans.  x  =  0,  tt---. 

3.  y  =  6  x^  —  x^.  Ans.  x  =  \.  ^  ' 

4.  ?/  =  sin  X.     Ans.  x  =  0,  tt,  etc.         7.   y  =  (logx)^.  Ans.  x  =  e^. 

Art.  3.    Exceptional  cases;    general  theory.     The  exceptional  case 
that  both  the  first  and  the  second  derivative  vanish  for  the   abscissa 
which  we  are  examining,  will  be  treated  best  by  developing  the  criteria 
for  maxima  and  minima  from  the  definition  in  a  more  general  manner. 
The  definition  may  be  expressed  in  symbols  as  follows : 
The  differences      .  -  .     ,  , 

f(x  +  h)  -f(x)  and  f(x  -  h)  --fix) 

have  like  signs  at  a  maximum  or  minimum  (negative  at  a  maximum 
and  positive  at  a  minimum),  while  they  have  unlike  signs  at  a  point 
which  is  neither  a  maximum  nor  a  minimum. 
By  Taylor's  Theorem, 

(1)  f{x  +  h)  -fix)  =  hfix)  +  ^f"(x)  +  ^f"'(x)  +  ^/'v(:r)  +  .... 

(2)  fix  -  h)  -fix)  =  -  hfix)  +  f^f'ix)  - 1^/"'(^)  +  ^/"W  -  -. 

We  now  apply  the  following  theorem : 

For  an  mjinite  series  of  increasing  positive  integral  powers  of  some  variable^ 
as  h  {convergent  for  the  values  ofh  to  he  used),  there  exists  a  positive  number 
//,  such  that  for  all  values  of  h  numerically  less  than  H,  the  first  term  of  the 
series  is  numerically  greater  than  the  aggregate  of  the  others. 

(If  we  take  a  very  small  number  as  H,  the  theorem  seems  plausible, 
since  the  higher  powers  of  H  would  be  exceedingly  small  in  comparison 
with  the  lowest  power  of  H  occurring.     For  a  proof  of  the  theorem  we 
refer  to  works  on  algebra.) 
35 


368  CALCULUS  [Ch.  XI. 

By  means  of  this  theorem  we  see  that  if /'(:c)  is  not  zero  for  the  value 
of  X  in  question,  then  for  sufficiently  small  values  of  h,  the  signs  of  (1) 
and  (2)  will  be  determined  by  the  first  terms  respectively,  and  will  hence 
be  unlike. 

Introducing,  for  this  discussion  only,  the  abbreviation  Mm.  for  "Maxi- 
mum or  Minimum,"  we  have  just  seen  that  there  can  be  no  Mm.  for  any 
value  of  X,  for  which  f'{x)  ^  0,  and  we  have  as  a  necessary,  though  not 
sufficient,  condition  for  a  Mm.  at  x,  f  {x)  =  0. 

If  f'(x)  =  0,  the  two  series  will  begin  with  the  terms  in  h'^.  If 
f"(x)  ^  0,  the  signs  of  the  series  will  be  like,  and  there  is  a  Mm. 

If,  however, /"(x)  =  0  also,  the  series  will  begin  with  the  terms  in  A^. 
If  f"'{x)  ^  0,  the  signs  will  be  unlike  when  h  is  sufficiently  small,  and 
there  is  no  Mm.  If  f"'{x)  —  0  likewise,  but  /'^"{x)  ^  0,  there  is  a  Mm. 
Proceeding  in  this  way,  we  reach  the  following  general  result : 

If  the  rth  derivative,  f^'^^x),  is  the  first  of  the  successive  derivatives  which 
does  not  become  zero  for  the  value  of  x  in  question,  then  if  r  is  odd,  x  does 
not  correspond  to  a  Mm. ;  hut  if  r  is  even,  there  is  a  Mm.  for  this  value  of 
X,  a  maximum  if  f^^\x^  is  negative,  and  a  minimum  if  it  is  positive. 

It  follows  readily  that  when  r  is  odd,  and  greater  than  unity,  x  is  the 
abscissa  of  a  point  of  inflexion. 

Art.  4.  Collected  criteria  concerning  forms  of  curves.     We 

collect  into  a  table  the  principal  results  which  we  have 
established  relative  to  the  forms  of  curves  for  a  given 
abscissa.     We  consider  the  curve  whose  equation  is 

y  =/(^), 

and  the  table  indicates  what  is  the  shape  of  the  curve,  if 
for  any  given  abscissa,  the  various  derivatives  assume  the 
values  specified.  With  the  exception  of  the  first,  the  condi- 
tions indicated  are  sufficient  but  not  necessary.  The  marks, 
•  ••,  mean  that  the  value  which  the  derivative  may  assume  is 
immaterial.  In  the  table,  as  throughout  our  treatment,  we 
exclude  the  case  that  any  of  the  derivatives  become  infinite, 
for  the  value  of  x  in  hand. 


3-5.] 


MAXIMA  AND  MINIMA 


369 


TABLE    OF    CRITERIA    FOR   CURVES 


au 

d'p 

dhj 

d^y 

dac 

aao^ 

diJC^ 

dx^ 

1. 

0 

... 

... 

... 

Tangent  parallel  to  a?-axis. 

2. 

0 

— 

... 

... 

Maximum. 

3. 

0 

+ 

... 

... 

Minimum. 

4. 

••• 

0 

^0 

... 

Point  of  inflexion. 

5. 

0 

0 

0 

— 

Maximum. 

6. 

0 

0 

0 

+ 

Minimum. 

7. 

^0 

+ 

•  •• 

... 

Convex  downward. 

8. 

^0 

— 

... 

.  • . 

Concave  downward. 

Art.  5.   Examples  of  maxima  and  minima.     I.    To  divide 
a  given  length  a  into  two  segments  x  and  a  —  x^  whose  product 
shall  he'  as  great  as  possible.     Let  y  denote  this  product,  so 
that 
(1)  y  =  x(^a-x^. 

Forming  the  derivatives,  we  have 


(2) 


dx  dx^ 


d% 


and  putting  -^  equal  to  zero,  we  find 
dx 


(3) 


0  =  a  ~  2  a:^,    or   x 


-2, 


That  is,  at  the  point  whose  abscissa  is  -,  -f-  is  zero,  while  —4 

^  dx  dx^ 

is  not  zero,  but  is  negative,  and  we  have  a  maximum. 


II.    Two  sides  of  a  triangle  being  given,  to  find  the  included 
angle  for  which  the  area  of  the  triangle  is  a  maximum. 


370  CALCULUS  [Ch.  XI. 

Let  the  two  sides  be  denoted  by  a  and  &,  the  angle  by  x^ 
and  the  area  by  y.     Then,* 

(4)  y  =z  ^ah  sin  x. 
The  first  derivative  is 

(5)  -^  =  J  ah  cos  X ; 
ax 

hence  at  a  maximum, 

(6)  ^  a6  cos  a;  =  0 ;   whence  cos  a:  =0,    or   x  =  -* 
The  second  derivative  is 

(7)  — ^  =  —  I  aft  sin  2: ;  » 

and,  since  sin  -^  =  1»  the  value  of  the  second  derivative  at 
the  point  whose  abscissa  is  — ,  is 


X 


=  —  ah^ 


and  a  maximum  occurs  at  that  point. 

Therefore,  the  triangle  will  have  the  maximum  area  when 
the  included  angle  is  a  right  angle. 

III.  To  find  the  base  and  altitude  which  a  rectangle  of 
given  area  A  must  have  in  order  that  the  perimeter  P  shall 
he  a  minimum. 

If  the  base  be  denoted  by  x^  then  the  altitude  is  — ,  and 

X 

the  perimeter  is 

(8)  P  =  2  a:  +  2  .  ^  =  2  f a:  +  -\ 

X         \        xj 


*  Formula  55,  Appendix. 


5.] 


MAXIMA  AND  MINIMA 


371 


Differentiating,  and  putting  the  first  derivative  equal  to 
zero,  we  have 

(9)  K^-f}='^ 

or,  x^  =  A^   X  —  -VA. 

Forming  the  second  derivative  we  find  that  it  is  negative 
for  X  =  V^,  and  therefore  the  square  has  the  least  perimeter 
of  all  rectangles  of  the  same  area. 

IV.  To  find  the  shortest  straight  line  that  can  he  drawn 
through  a  given  point  A^  within  a  right  angle^  to  the  sides 
of  the  right  angle  (Fig.  64).  g 

Let   the    required   line    be   de- 
rioted  by  BO^  the  coordinates  of 
the   point   BE  =  a   and   AE  =  h, 
and  the  lines  BQ  and  BB  by  a:     D 
and  y^  respectively. 

From  the  two  similar  triangles  BBC  and  AEO  we  have 

the  proportion 

y  :  X  ::h  :  (x  —  a^, 

bx     ^ 


whence 


y 


X  —  a 


and 


BC^  =  x^. 


hH^ 


(10) 


The  first  derivative  is 
d(B^^ 


(x  —  ay 


2x 


(X 


ay  '2Px-  y^x^  '  2(x  -  g)  , 
dx  ~ "   '       .  C^  —  ^)* 

and  at  a  minimum  we  must  have 


(11)         2x  I  (^-^y-252:.-^2^2.2(^-a)_^^ 

(^x  —  ay 


or 


2xl(ix- ay -ab^]=0. 


372  CALCULUS  [Ch.  XL 

Hence,  by  solving  for  x,  either 

a;  =  0,    or   x  =  a  -\-  waP^ 

and  the  position  of  the  point  C  for  a  line  BO^  which  may  be 
a  minimum,  is  thus  determined  in  terms  of  the  coordinates 
of  the  given  point. 

We  leave  the  examination  of  the  second  derivative  as 
an  exercise  for  the  student.  Each  value  proves  to  be  a 
minimum. 

V.  What  sector  of  a  given  circle  will  form  the  convex  sur- 
face of  a  cone  of  maximum  volume  ^ 

Let  the  radius  of  the  given  circle 
be  r  (Fig.  65}^  and  let  the  angle  of 
the  sector  forming  the  surface  of  the 
cone  be  (f)  (in  circular  measure).  The 
arc  of  the  sector  is  then  (^r,  and  this 
is  the  perimeter  of  the  base  of  the 
cone.  Denoting  the  radius  of  the 
Fig.  65.  base  of  the  cone  by  jR,  we  have 

27rE  =  cl>r,  or  E  =  i^' 

If  h  be  the  altitude  of  the  cone,  its  volume  is 


(12) 


V: 


IT 


mh 


Since   ^,  i^,  and  the  slant  height  r  of  the  cone  form  a 
right-angled  triangle, 


y^  =  Vr2-i^ 


=  r\'. 


4  7r2 


*  Formula  67,  Appendix. 


5.]  MAXIMA  AND  MINIMA  378 

Substituting  in  equation  (12),  we  have 


(14)  V=^  ^ Vl  --^  =  ^Vl  -  ^, 

and  we  now  have  to  find  what  value  of  (/>  makes  V  a  maxi- 
mum.    Any  value   of  </>  which  makes   V  a  maximum  will 

also  make  — ^  times  V  a  maximum  ;  that  is, 


(15)  ^.  =  ^^=Wl-4$ 

will  be  a  maximum  whenever  V  is  such.  It  is  therefore 
sufficient  to  examine  V^  for  maxima  or  minima.*  The 
equation  of  condition  for  <^  is 

^        4  7r2 


Multiplying  through  by  \/l  —  -^—z  (this  is  never  zero,  for 

4  TT^ 

</)  manifestly  cannot  have  the  value  2  tt),  we  obtain 


(17) 

2,^(1- 

4  7rV 

4  7r2 

Dividing  by  </>, 

we  have 

2- 

3  <!>' 

4,r2 

=  0, 

or 

(18) 

<i> 

=  2,rV|. 

0. 


*  In  general,  C  being  any  constant,  the  maxima  and  minima  of  Of(x)  are 
the  same  as  those  of /(x).  For  in  the  one  case  we  have  to  solve  the  equation 
Cf'(x)  =0  and  in  the  other /'(a:)  =  0,  and  both  these  equations  have  the 
same  roots. 


S74  OALcuim  [ch.  XI. 

The  corresponding  angle  in  degrees  is  approximately 
X  =  294°. 

The  volume  of  the  maximum  cone  is 

(19)  F=?f^VI. 

That  the  value  of  <f>  actually  corresponds  to  a  maximum  may  be  seen 
as  follows  without  examining  the  set3ond  derivative.  If  (f>  =  2  ir,  the 
cone  is  'the  circle  itself  and  its  volume  is  zero;  if  <l>  =  0,  the  cone  is  a 
straight  line  (the  radius)  and  its  volume  is  likewise  zero.  Between  these 
two  zero  volumes  there  must  be  at  least  one  maximum,  and  as  the  first 
derivative  vanishes  for  only  one  value  of  cf>  between  zero  and  2  tt,  that 
value  determines  the  maximum. 

VI.  The  following  is  an  example  of  a  function  whose 
second  derivative  vanishes  simultaneously  Avith  the  first 
derivative : 

(20)  i/  =  a^-Sx'^-{-nx  +  2. 

The  first  and  second  derivatives  are,  respectively, 

^  =  Sx^-6x-i-S 
ax 

and  — ^  =  6  x  —  Q. 

dx^ 

The  roots  of  the  equation 

(21)  3  2^2 -62;  + 3  =  0 

both  are  unity  and  the  corresponding  ordinate  is  ^  =  3.  In 
the  point  whose  coordinates  are  x  =  \  and  ^  =  3,  the  tangent 
of  the  curve  is  parallel  to  the  aj-axis.  When  x  =  l^  however, 
the  second  derivative  becomes  equal  to  zero  ;  and  the  point 
in  question  does  not  present  a  maximum  or  minimum,  but 
rather  a  point  of  inflexion. 

Art.  6.   Minimum  of  intensity  of  heat.     Let  A  and  B  he 

two  point-sources  of  heat.     It  is  required  to  find  the  point  M 


ft 


I]  MAXIMA  AND  MtHtMA  875 


on  the  stnaight  line  AB^  which  is  at  the  lowest  temperature^ 
the  intensity/  of  the  radiation  of  heat  varying  inversely  as  the 
square   of  the   distance  from  the 

source  of  heat.  T jjj g 

Let  d   represent   the    distance  Fig.  66. 

between    the    points    ^    and    ^ 

(Fig.  66^^  and  x  the  distance  from  A  of  the  point  M  on  the 
straight  line ;  then 

(1)  MA  =  x  and   MB  =  d  -  x. 

Let  the  intensities  of  the  heat  at  unit  distance  from  the 
sources  of  lieat  be  denoted  by  a  and  yS,  respectively.  The 
total  intensity  of  heat  o)  at  the  point  M  is 

(2)  <o=^+       ^      , 

x^       (c?  —  2^)2 
and  we  wish  to  find  for  what  values  of  x  this  expression  is  a 
minimum.     We  have 

d(d^_2a  20 

dx  7? .      (d  —  x^^ 

and  the  points  we  seek  are  determined  by  the  equation 

x^       (c?  —  x^^ 

whence  jd-xY^fi^ 

x^  a 


By  extracting  the  cube  root  of  both  members  we  obtain 
X  d  —  X  _  VyS 


The  distances  BM  and  AM  liave,  therefore,  the  same 
ratio  as  the  cube  roots  of  the  corresponding  heat  intensities. 
By  solving  the  last  equation  we  liave,  further, 

a^d 


(4) 


«»  + 


376 


CALCULUS 


[Ch.  XI. 


In  this  case  it  is  necessary  to  ascertain  whether  the  value 
found  corresponds  to  a  maximum  or  a  minimum.  This  is 
easily  done  by  means  of  the  second  derivative, 


(5) 


2.3a       2.3^ 


which  is  positive  for  all  values  of  x^  inasmuch  as  the  powers 
of  x  and  d  —  x  are  even,  and  a  and  ^  are  essentially  positive. 

Art.  7.  The  law  of  reflection.  It  is  required  to  find  a 
point  so  situated  on  the  straight  line  GH  (Fig.  67)  that  the 
sum  of  its  distances  from  two  fixed  points  A  and  B  is  a 
minimum. 

Let  the  perpendiculars  AA'  and  BB'  be  designated  by  a 
and  5,  respectively;  also  let  A^ B'  =  p.  P  being  any  arbi- 
trary point  between  A'  and  B\  let 
its  distance  from  A'  be  denoted  by 
rr,  so  that  PB'  =  p  —  x.  From  the 
two  right  triangles  AA'P  and  BEP, 
we  have 

AP  =  Va2  +  x^ 

(1)  , 

BP=Vb^+{p-x)^ 

the  square  roots  being  taken  with 
the  positive  sign. 

Since  the  sum  of  AP  and  BP  is 
to  be  a  minimum,  we  have  to  deter- 
mine what  value  of  x  will  make 


Fig.  67. 


(2) 


f(x)  =  Va^  +  x^-\--Vb^+(ip- 
a  minimum. 

By  differentiation  we  have 

X         _        ( p  —  x) 


xy 


/'(^)= 


Va2  +  a^      V62  +  (^  _  xy 


6-7.]  MAXIMA  AND  MINIMA  377 

and,  since  this  must  be  equal  to  zero  at  a  maximum  or  mini- 
mum, we  have 
(3)  ^ = (P-^) 


Va2  +  x^      ^/P+(p-xy 

an  equation  which  leads,  by  squaring,  to  a  quadratic,  whose 
roots  are  readily  found  to  be 

ap     >         ap 
X  =      ^      or  -—^ — 
0  —  a         0  -\-  a 

Of  these  values  only  the  latter  satisfies  equation  (3),  the 
former  having  been  brought  in  by  squaring. 

In  order  to  ascertain  whether  a  minimum  actually  exists 
we  examine  the  second  derivative, 

(4)  f'Cx)  = ^  + ^- ; 

this  is  positive  for  all  values  of  x^  showing  that  the  value  of 
X  found  from  (3)  determines  a  minimum. 

Equation  (3)  yields  a  simple  geometric  result.  If  we 
denote  the  angles  APA'  and  BPB'  by  <f>  and  i/r,  we  see 
from  equation  (3)  that 

cos  0  =  cos  i/r  ; 

the  minimum  occurs  at  the  point  where  the  lines  AP  and 
BP  make  equal  angles  with  the  given  straight  line.  If  we 
conceive  A  to  be  a  source  of  light  and  GiH  a  reflecting 
surface,  it  is  known  that  the  ray  of  light  AP  will  be 
reflected  in  the  direction  PB  such  that  (f)  =  yjr;  i.e.  light 
which  travels  from  A  to  ^  by  reflection  from  CrH  takes  a 
path  which  is  a  minimum. 

It  is  easily  seen  that  if  the  point  A^  be  so  taken  in  the 
straight  line  AA'^  and  on  the  opposite  side  of  GrH  that 
A^A'  =  A' A,  then  the  points  A^PB  lie  in  a  straight  line. 


378 


CALCULUS 


[Ch.  XI. 


Art.  8.  The  law  of  refraction.  Two  points  A  and  B  lie 
on  opposite  sides  of  a  straight  line  GrH.  If  a  point  moves 
from  A  to  B  in  the  shortest  tirne^  and  if  its  velocity  is  uniform 
hut  different  on  each  side  of  the  line,  where  does  it  cross  the 
line  f 

We  drop  perpendiculars  from  A  and  B  to  GH,  and  let 
AA!  =  a,   BB'  =  h,  and  A'B'  =p.     Further,  let  F  be  any 

point  on  GH  between  A'  and 
B',  let  X  denote  its  distance 
A'P  from  J.^  and  p  —  x  its 
distance  from  B' .  Let  V^ 
and  V^  be  the  velocities  per 
second  above  and  below  GH 
respectively.  Then  the  path 
APB  is  traversed  in  t  seconds, 
where 

AJP^^BP 


t 

fv 

L 

'\ 

a 

\ 

c. 

\ 

P 

»' , 

^  / 

<' 

\ 

•\ 

V 

b 

M 

\ 

N 

B 

Fig.  68. 


(1)      t  =  ^ 


This  is  the  expression  which  is  to.be  examined  for  a  mini- 
mum.    In  the  triangles  APA'  and  BPB'  we  see  that 


(2) 
hence, 


AP  =  Va2  +  x^  and  BP  =  VP  +  (p  -  xj^ 


(3) 


Va^  +  a;^  ,   ^p^(p-xy 


Vi 


dt 


Applying  the  criterion,  we  must  have  —  =  0,  or 

(XX 


(4) 


(y^-^) 


V^^a^  +  x^      V^-\/b'^  +  {p-xy 


=  0 


This  leads  to  an    equation   of   the   fourth   degree   when 
rationalized;    but  we  can  simplify  matters  by  certain  geo- 


MAXIMA   AND  MINIMA  379 

metric  considerations.  If  at  P  we  erect  a  perpendicular 
LM  to  GIT,  and  denote  the  angles  AFL  and  BFM  by 
<^,and  i/r,  respectively,  we  have 

-  =  sm  (p,     —     ^  =  sm  y. 

Substituting  these  values,  equation  (4)  becomes 
sin  (/)  _  sill  g/r 

or 

^  ^  sin  t/r      V^ 

There  can  be  no  minimum  except  for  points  between  A' 
and  B' ;  for  if  P  be  supposed  to  move  beyond  either  of 
these,  both  parts  of  the  patii  APB  will  constantly  increase 
as  P  moves  on,  and  therefore  the  time  of  traversing  the 
path  APB  will   also  increase. 

The  actual  solution  of  the  equation  resulting  from  equat- 
ing the  first  derivative  to  zero,  and  the  substitution  of  the 
value  thus  found  in  the  second  derivative,  may  often  be 
avoided  by  proving  ^from  the  geometrical  conditions  that 
a  minimum  or  a  maximum  must  exist.  In  the  present  case 
this  is  easily  done.  Taking  any  point  P  within  the  interval 
A' B'^  points  Pj  and  P^  beyond  B'  and  A\  respectively, 
exist  such  that  AP^B  and  AP^B  are  both  greater  than 
APB.  As  the  function  is  continuous,  and  does  not  become 
infinite  between  Pj  and  Pg,  there  must  be  a  minimum  some- 
where. We  have  already  noticed  that  no  minimum  can 
occur  except  in  A' B' ^  and  in  this  interval  there  is  only  one 
point  at  which  the  necessary  condition  for  a  maximum  or 
minimum  is  satisfied.  The  minimum  must  accordingly 
occur  at  this  point,  i.e.  at  the  point  P  where  the  straight 


380  CALCULUS  [Ch.  XI. 

lines  AP  and  BP  make  with  the  perpendicular  LM  angles 
whose  sines  have  the  same  ratio  as  the  corresponding 
velocities.  The  actual  determination  of  the  position  of  P  is 
of  no  special  interest  in  this  conixection. 

We  now  assume  that  ^^  separates  two  media  of  different 
composition.  With  each  medium  there  is  connected  a  con- 
stant, which  is  inversely  proportional  to  the  velocity  of  the 
propagation  of  light  in  the  medium,  and  which  is  known  as 
the  index  of  refraction.  (The  standard  of  comparison,  to 
which  the  index  unity  is  assigned,  is  the  vacuum.)  If  a 
ray  of  ligtit  passes  from  A  to  B^  then  the  path  taken  by  the 
ray  is,  in  accordance  with  the  Law  of  Refraction,  such  that 
the  incident  and  refracted  rays,  i.e.  AP  and  BP.,  make 
with  the  normal  (^LM)  angles  (<^  and  i/r)^  whose  sines  are 
inversely  proportional  to  the  indices  of  refraction ;  that  is, 
inversely  proportional  to  the  velocities  of  the  propagation 
of  light  in  the  media,  so  that  the  above  equation  represents 
the  Law  of  Refraction.  The  ray  is  accordingly  refracted 
so  that  the  path  AB  is  traversed  in  the  shortest  time. 

EXERCISES   XL 

Examine  the  following  for  maxima  and  minima : 

7x  +  6 


1.   y  = 


x-lO 


2. 

y  =  ^  +  ^  (a>0). 

3. 

y  =  X  -{-Vl  —  X. 

4. 

y  =  x  +  -' 

5. 

y  =  x-. 

6. 

1 

y  =  :^. 

Ans.  min.  a;  =  a  ;  max.  x  ^=  —  a 
A  ns.  max.  x  =  ^ 

Ans.  min.  a:  =  +  1 ;  max.  x  =  —  1 

.  1 

Ans.  max.  x  =  - 

e 

Ans.  max.  x  —  e 
1.   y  =  x^  +  2px  +  q.  *  Ans.  min.  x  =  —  L. 


10.   y  =  xP(a  —  xy(a^O;   j9,  ^,  +  integers).    Ans. 


11 

y 

X 

1  +  x^ 

12. 

y 
11 

_{x  +  3)3 
{x  +  2y 

1  -  ;r  +  a:2 

MAXIMA  AND  MINIMA  381 

8.   y  =(x  —  \){x  —  2)(x  —  3^.     ^n6\  max.  a:  =  2 -\  niin.  a:  =  2H -. 

V'8  V3 

g    , ,  _  _^.  yl7i.s.  max.  x  =  e. 

^    ^  X 

'  min.  a:  =  0  if  jo  even, 
min.  x  =  (i  if  q  even. 

pa 
max.  a:  =  — ; — 
p  +  q 

aq 

max.  x  =  a ; — 

p-^q 

A  \  max.  X  =  1. 

Ans.    I     . 

(  mm.  :p  =  —  1. 


Ans.  min.  a:  =  0. 


^ns.  min.  x  =  ^. 
1  +  x  —  x'^ 

14.  2/ =  sina:(l  +  cosx).  ^/is.  x  =  ^. 

15.  What  fraction  exceeds  its  square  by  the  greatest  number  possible  ? 

Ans.  I. 

16.  What  rectangle  of  given  perimeter  has  the  largest  area  ? 

Ans.  The  square. 

17.  Divide  10  into  two  such  parts  that  the  product  of  their  cubes 
shall  be  as  great  as  possible.  Ans.  5,  5. 

18.  Given  a  cylindrical  tree  trunk  of  diameter  D  and  of  sufficient 
length  ;  to  cut  from  it  a  rectangular  beam  which  shall  have  the  greatest 
strength,  given  (from  the  theory  of  strains),  that  the  resistance  of  a 
rectangular  beam  is  directly  proportional  to  bh^,  where  ?>  and  h  are  the 
dimensions  of  the  section  of  the  beam,  the  pressure  being  applied  per- 
pendicularly to  the  side  of  breadth  b.  Ah—  ^^^ -  h  —  ^^ 

19.  A  submarine  telegraphic  cable  consists  of  a  central  circular  part 
called  the  core,  surrounded  by  a  concentric  circular  part  called  the  covering. 
K  X  denote  the  ratio  of  the  radius  of  the  core  to  that  of  the  covering,  it 

is  known  that  the  speed  of  signaling  varies  as  a^^log—     Show  that  the 

1  ^ 

greatest  speed  is  attained  when  x  = 

Ve 


382  CALCULUS  [Ch.  XL 

20.  From  the  corners  of  a  given  rectangnlar  piece  of  tin  (dimensions 
a  and  b)  square  pieces  are  to  be  cut,  so  that  the  pan  formed  by  turning 
up  the  sides  so  produced  shall  have  as  great  a  volume  as  possible. 

Ans,  Side  of  squares  cut  out  =  a  +  b  -  Va^  -  ab±b^ 

21.  What  rectangle  of  given  area  has  the  smallest  perimeter  ? 

Ans.  The  square. 

22.  In  a  horizontal  plane  the  distance  d  between  two  points  A  and  B 
is  known.  Given,  that  the  intensity  of  light  varies  directly  as  the  sine  of 
the  angle  of  incidence ;  and,  inversely,  as  the  square  of  the  distance,  to 
find  at  what  height  h,  perpendicularly  above  A,  an  incandescent  point 
must  be  situated  in  order  that  the  intensity  of  light  from  it  at  the  point 
B  may  be  at  a  maximum.  .        ,       r/V2 

2 

23.  An  open  bin,  with  square  base  and  vertical  sides,  is  to  hold  a  given 
volume  of  wheat.  What  must  be  its  inner  dimensions  in  order  that  as 
little  material  as  possible  may  be  needed  to  construct  it,  the  thickness  of 
the  material  being  disregarded  ? 

Ans.  The  depth  must  be  half  the  width. 

24.  An  object,  AB,  of  length  /  is  a  perpendicular  to  a  given  plane, 
and  the  lower  end  of  AB  is  at  the  distance  d  from  the  plane.  At  what 
distance  x  from  the  point  where  A  B  produced  meets  the  plane  must  an 
observer  in  the  plane  stand  in  order  that  the  object  may  appear  the 
largest?  Ans.  x  =  y/d{d  +  /). 

25.  To  inscribe  the  largest  possible  rectangle  in  a  given  triangle. 
Ans.   The  altitude  of  the  rectangle  must  be  half  the  altitude  of  the 

triangle. 

26.  A  person  in  a  boat  three  miles  from  the  nearest  point  P  of  a 
straight  beach  wishes  to  reach  in  the  shortest  time  a  place  on  the  shore 
five  miles  from  the  point  P;  if  he  can  run  five  miles  per  hour,  but  row 
only  four  miles  per  hour,  at  what  place  must  he  land? 

Ans.  One  mile  from  the  point  to  be  reached. 

27.  A  body  is  projected  upwards  at  angle  a  with  velocity  c  ;  to  what 
height  will  it  attain,  disregarding  the  resistance  of  the  air? 

Hint.    We  know  from  physics  that  under  the  given  conditions  the 

body  will  attain,  in  x  seconds,  the  height 

qx^ 
ex  sm  a  —  ^, 

g  being  the  constant  of  gravity.    This  is  to  be  a  maximum.  Ans. 


8-9.] 


MAXIMA  AND  MINIMA 


383 


28.  A  perpendicular  lamp  post  is  to  be  erected  at  a  given  horizontal 
distance  d  from  a  statue.  What  must  be  the  height  of  the  lamp  above 
the  head  of  the  statue  in  order  that  the  top  of  the  head  may  be  most 
strongly  illuminated? 

Hint.   According  to  physics,  the  intensity  of     ^ 
the  illumination  is  inversely  proportional  to  the 
square  of  the  distance  of  the  light  from  the  illu*^' 
minated  point,  and  directly  proportional  to  the 
sine   of   the  angle    at   which  the  rays   of   light 


sin  LSP 

PS  =  (/, 


Fig.  69. 


strike  the  object.     We  have  then,  that 
has  to  be  a  maximum,  or,  if  LP  =  x, 

maximum.     We  may  simplify  this  by  squaring  and  putting 
when  the  expression  to  be  a  maximum  becomes 

z^  -  dh.^. 


is  to  be  a 
1 


(/2+x2 


Ans.  X  =  —iJo 

29.   At  what  height  on  a  perpendicular  wall  must  letters  of  a  given 
height,  h,  be  placed  in  order  to  appear  the  largest  to  a  spectator  at  a  given 

distance  d  ? 

Hint.    Let    AB  =  d,    CX  =  h,   BX  =  x. 
Angle  CAX  is  to  be  maximum. 
We  have 

rf2  +  a:2  +  hx 


cot  CAX  ^  cot  (CAB  -  XAB)  = 


hd 


This  has  to  be  minimum  in  order  that  angle 
may  be  maximum. 
h 


Fig.  70. 


A  71S.     X  = 


i.e.  the  middle   of  letters 


must  be  on  a  horizontal  line  with  eye. 


30.  What  is  the  minimum  of  material  (disregarding  the  thickness) 
needed  to  make  a  right  cylindrical  vessel,  open  at  the  top,  of  given 
volume  F?  Ans.  sV^iTV^^ 

31.  What  is  the  minimum  of  material  needed  in  the  previous  problem 
i  f  the  thickness,  c,  be  taken  into  account  ? 

Ans.  3  cy/^iy^^  +  3  c^</^irW  +  ttc^. 


Art.  9.   Estimation  of  errors.     It  is  not  often  possible  or 
practicable  to  measure  a  given  quantity  directly.     Usually 
20 


384  CALCULUS  [Ch.  XI. 

one  or  more  quantities  are  measured  which  stand  in  a  known 
relationship  to  the  required  one.  Thus,  the  equivalent 
weight  of  sodium  is  not  determined  directly  by  preparing 
sodium  chloride  out  of  weighed  amounts  of  sodium  and 
chlorine,  or  by  decomposing  a  known  weight  of  sodium 
chloride  into  its  components,  but  indirectly  by  finding  what 
amount  of  silver  can  replace  the  sodium,  and  by  calculating 
the  required  equivalent  weight  on  the  assumption  that  the 
composition  of  silver  chloride  is  known;  likewise,  in  the 
measurement  of  resistance  by  means  of  a  Wheatstone  Bridge 
the  required  quantity  is  not  found  directly  but  indirectly 
by  calculation  from  the  relationships  of  the  urms  of  the 
Bridge;  likewise,  the  reading  of  a  galvanometer  deflection 
does  not  give  at  once  the  strength  of  an  electric  current, 
but  this  is  found  only  after  the  trigonometric  tangent  of  the 
angle  of  deflection  is  known,  etc.,  etc. 

In  all  such  cases  as  these,  the  methods  of  the  present  chap- 
ter are  of  great  assistance  in  determining  the  best  mode  of 
conducting  the  experiments  and  of  criticising  the  results. 
Let  ^  be  a  required  quantity  to  be  determined  from  another 
quantity  x  directly  measurable,  and  known  to  be  connected 
with  y  by  the  relation 

Mathematically,  the  quantity  y  is  hereby  fixed  as  soon  as 
X  is  known ;  but  under  our  assumption  x  is  to  be  determined 
experimentally,  and  accordingly  can  never  be  known  accu- 
rately, but  only  approximately. 

Each  observed  value  of  x  will  differ  by  some  error  Arr 
from  the  true  value.  This  error  ^x  is  of  course  not  known, 
except  that  it  lies  between  certain  limits.  Nevertheless,  it 
has  an  influence  upon  the  calculated  value  of  y^  causing  the 


9.]  MAXIMA  AND  MINIMA  385 

latter  to  differ  from  the  true  value  by  an  error  which  we  call 
Ay.     The  error  in  the  final  result  accordingly  amounts  to 

(1)  At/=fCx  +  Ax)-f(x), 

where  x  denotes  the  observed  value  and  x  +  Ax  the  true 
value. 

By  Taylor's  Theorem, 

f(^x  +  Ax)=f(x')-^fCx')Ax+:t^Ax^  +  ..., 

Ax  must  certainly  be  a  very  small  quantity,  or  the  measure- 
ment itself  is  comparatively  worthless.  Hence,  in  the 
expansion  into  a  Taylor's  Series,  we  may  omit  all  terms  after 
the  second,  obtaining  as  a  close  approximation, 

(2)  Ai/=fix}Ax, 
and  as  the  relative  error^* 

(3)  ^=/:ma:.. 

Equation  (3)  finds  frequent  applications  in  the  critical 
revision  of  the  results  obtained  according  to  different 
methods  of  experimentation. 

EXAMPLES 

1.  In  determining  the  equivalent  weight  of  sodium  (p.  384)  it  has 
been  found  that  x  parts  of  sodium  chloride  are  precipitated  by  one 
part  of  dissolved  silver  (as  silver  chloride).  Let  A  be  the  equivalent 
weight  of  silver,  B  that  of  chlorine  (both  of  which  are  assumed  to  be 

■ — .^ 

*  Evidently  it  is  not  the  absolute  but  the  relative  error  which  determines 
the  accuracy  of  a  measurement.  When  we  determine  a  weight  to  within 
0.1  gram,  the  accuracy  attained  may  in  certain  cases  be  very  great,  while  in 
others  it  is  entirely  insufficient,  since  all  depends  upon  whether  0. 1  gram  is 
an  extremely  small  or  a  rather  large  fraction  of  the  total  weight.  (Soe 
p.  357.)  Multiplying  the  relative  error  by  100,  we  obtain  the  percentage  of 
error. 


386  CALCULUS  [Ch.  XI. 

known).     It  is  further  known  that  the  required  equivalent  weight  y  of 
the  sodium  may  be  found  by  means  of  the  equation 

(y  -\-  B)  :  A  =  x'A,   or   y  =  Ax  —  B. 

The  relative  error  is,  according  to  equation  (3), 


^  =  f!MAx  =  -A. 


Aa:, 


y        f{x)  Ax-  B 

or  ^^A/^:,  =  y±Jl^. 

y      y  y     X 

For  the  example  in  hand,  3/  =  23,  5  =  35.5,  approximately,  and 

.y  +  ^^23  +  35.5^og,^. 
y  23 

so  that  an  error  of  1  per  cent  in  the  determination  of  x  produces  an  error 
of  about  2.5  per  cent  in  y.  It  is  a  disadvantage  to  have  B  considerably 
larger  than  y.  In  the  case  of  barium  chloride  where  y  =  i|^  (equivalent 
weight  of  barium), 

2^+^  =  1.52. 

y 

Hence,  an  equivalent  weight  determination  of  barium  by  precipitation 
of  barium  chloride  with  dissolved  silver  gives  better  results,  other  condi- 
tions being  the  same,  than  does  that  of  sodium. 

2.  Let  X  denote  the  extent  of  a  chemical  reaction  at  the  time  t. 
The  speed  of  reaction  y  (previously  represented  by  ¥)  may  be  found 
according  to  the  considerations  on  p.  240  et  seq.  by  means  of  the  general 
formula 

y  =  -  <^(^)» 

where  ^(a:)  denotes  a  function  depending  upon  the  nature  of  the  reac- 
tion in  question  ;  t  as  well  as  x  must  be  measured  in  order  to  calculate  y, 
but  as  a  rule  t  can  be  determined  so  closely  that  there  is  no  appreciable 
error,  while  x  is  affected  by  the  error  Aa:.     We  therefore  have 

Az/=-<^'(a:)Aar. 

When  the  number  of  observations  of  the  same  phenomenon  is  large 
(as  in  Chap.  VII,  p.  244),  the  values  of  Aa:  may  be  taken  as  equal,  and  the 
error  of  y  calculated  from  each  value  of  x  may  be  put  proportional  to 
T-SEl^  and  the  reliability  of  this  value  of  y  proportional  to  ♦    In 

t  4>\x) 


9.]  MAXIMA  AND  MINIMA  887 

such  cases  it  is  not  permissible  to  take  the  mean  of  the  different  values 
found  for  y,  but  each  value  of  y  is  to  be  multiplied  by  the  corresponding 

value  of  (called  the  weight  of  the  observation),  and  divided  by 

the  sum  of  all  the  weights.     We  have  accordingly  the  formula 


in  which  each  observation  does  not  now  exert  equal  (uncriticised)  influ- 
ence, but  has  an  effect  on  the  result  determined  according  to  its  relia- 
bility. It  is  apparent  that  in  this  case  it  is  a  matter  of  indifference 
whether  we  carry  out  our  computations  with  the  relative  error  — ^,  or 
more  simply  as  we  have  done  above,  with  the  error  Ay  itself.  ^ 

According  to  equation  (3)  the  relative  error  is  dependent  upon  Ax, 

the  error  of  observation,  and  upon   the  quantity,   "^t^t^-     Both  these 

factors  are,  therefore,  to  be  made  as  small  as  possible.  This  is  accom- 
plished in  the  case  of  Ax  by  making  our  measurements  as  accurate  as 

f'(x) 
possible,  and  in  the  case  of  \.}^  by  so  arranging  the  experiment  that 

this  fraction  becomes  a  minimum. 

The  last  condition  is  fulfilled  (p.  364)  when  the  derivative 

^r/'(^)i^o. 


^  ^  dxlf(x).\ 


It  is  often  impracticable  to  arrange  the  experiment  so  that  this  condi- 
tion shall  be  fulfilled.  How  it  is  achieved,  when  practicable,  will  be 
illustrated  in  the  following  examples: 

3.  Measurement  of  resistance  by  a  Wheatstone  Bridge.  The  required 
resistance  y  is  computed  from  the  formula 

(5)  y  =f(x)  =  w  ^, 

where  w  is  the  compensating  resistance,  I  the  length  of  the  slide-wire, 
and  a:  the  position  when  a  balance  is  secured.     In  this  case 

„,  .                 I           f(x)            I 
f(x)  =  w 5     '  \  ^  = , 

and  finally, 


dxlf(x)J         x\l-xy 


388  CALCULUS  [Ch.  XI. 

This  expression  becomes  equal  to  zero  when  x  =-.     The  error  of 

balance  (for  instance,  0.1  mm.)  has  on  this  account  the  least  influence 
on  the  end-result  in  the  middle  portions  of  the  bridge-wire.  It  is  there- 
fore good  practice  to  alter  the  compensating  resistance  so  that  in  securing 
the  balance  only  the  middle  portions  of  the  wire  may  be  used. 

4.  Measurement  of  current  strength  with  a  tangent  galvanometer.  The 
required  current  strength  y  is  proportional  to  the  tangent  of  the  angle 
of  deflection  x',  hence  y  —f{x)  is  in  this  case, 

y  —  C  tan  x. 

Now  /.(.)=  ^,m  =  _L_ 

cos^  £     f\x)      sm  X  cos  x 


and  ArZMl  = 

dxVf{x)\ 


sin^  X  —  cos^  X 


sm^  X  COS''  X 


The  last  expression  vanishes  when  sin  x  =  cos  x ;  that  is,  for  an  angle 
of  45°.  The  error  of  reading  has  therefore  the  smallest  influence  on  the 
final  result  when  in  a  given  case  the  dimensions,  turns  of  wire,  etc.,  are 
so  chosen  that  a  deflection  of  45°  is  obtained. 


r 


CHAPTER   XII 

DIFFERENTIATION   AND    INTEGRATION   OF   FUNCTIONS 
FOUND    EMPIRICALLY 

Art.  1.  Differentiation.  When  by  direct  observation  cer- 
tain relationships  between  two  variable  quantities  have  been 
found,  it  is  customary  first  of  all  to  collect  the  results  of  the 
measurements  into  a  table.  We  then  endeavor  according 
to  circumstances  either  to  find  a  mathematical  expression 
(interpolation  formula)  that  will  enable  us  to  compute  with 
as  good  an  approximation  as  possible  the  values  of  one  quan- 
tity from  those  of  the  other,  or  we  try  to  make  the  relation- 
ships found  clearer  by  a  graphic  representation.  While  in 
many  cases,  moreover,  the  derivative  of  one  of  the  quantities 
with  respect  to  the  other  is  of  theoretic  importance,  its  direct 
determination  is  of  course  impossible,  because  our  instru- 
ments measure  only  with  a  certain  degree  of  accuracy,  and 
are  therefore  not  able  to  follow  by  measurement  beyond  a 
certain  point  the  value  of  the  ratio  of  quantities  which 
approach  the  limit  zero.  But  if  we  are  in  possession  of  a 
sufficiently  good  interpolation  formula,  its  differentiation 
will  give  the  required  result ;  *  or  if  on  the  other  hand  we 

*  Thus  Horstmann  {Berichte  der  deutschen  chemischen  GeseUschaft, 
Vol.  2,  p.  137  [1869]),  letting  p  denote  the  tension  of  dissociation  of  sal 
ammoniac,  and  t  the  temperature,  made  use  of  the  interpolation  formula 

\ogp  =  a  -\-  bA* 

(where  a,  6,  and  A  are  constants  whose  values  may  be  taken  from  a  table) 


to  find  the  value  of  the  derivative 

dt 


dp 
389 


390  CALCULUS  [Cii.  XII. 

have  secured  an  accurate  graphic  representation,  tangents 
drawn  at  the  desired  points  of  the  curve  determine  the 
derivatives  approximately  (p.  117).* 

Both  methods  have  their  shortcomings ;  the  first  one 
assumes  that  we  are  in  possession  of  a  good  interpolation 
formula,  which,  however,  we  cannot  obtain  at  all  in  many 
cases,  and  which  almost  always  necessitates  quite  a  little 
tedious  computation  ;  the  second  one  requires  unusual  skill 
in  drawing  to  attain  results  of  much  accuracy. 

There  is,  however,  a  third  method  permitting  of  the 
determination  of  the  approximate  value  of  the  derivative 
directly  from  a  table  of  experimental  results.  Let  f(x)  be 
the  function  in  question,  and  let  us  suppose  that  we  know 
its  value  for  values  of  the  variable  differing  by  the  same 
amount,  as 
(1)  rr,  x±}i^  x±2h,  a:  ±  3  A,   •••. 

Such  a  problem  arises,  for  example,  when  there  is  known  the  vapor 

tension  p  for  a  liquid  at  temperatures  0  that  differ  among  themselves  by 

the  same  number  of  degrees  (one  degree,  for  instance),  and  it  is  required 

dp 
to  find  the  derivative  -^  for  a  given  tension  p  =  p^. 

We  give  the  formula  at  once,  letting  its  proof  come  later. 

,,      dp^l\\  +  ^_,     1A^L,+A^^,      1  A^%  +  A^-_,...K 
^''^    dS     h\       2  6  2  30  2  )' 

where  the  quantities  \,  A_i,  ^"-i-,  •••  have  the  following 
meaning.     If  we  put 


*  This  was  the  second  way  in  which  Horstmann  (Liebig''s  Annalen, 
Ergangzungshand,  8,  p.  125  [1871-1872])  found  the  derivative  mentioned 
in  the  preceding  footnote. 


1.]  DIFFERENTIATION  OF  FUNCTIONS  891 

then  ^1-^0  =  ^0'  Po    -P-i  =  ^-v 

P2-Pi  =  \^    P-l-P-2  =  ^-2^ 

etc.,       etc. 
The  quantities 

Ai,  Aq,  A_i,  A_2,  - 

represent  the  differences  of  the  successive  values  of  the 
pressure  ;  we  call  it  the  first  series  of  differences.  Further- 
more, the  quantities 

Ai-Ao  =  AV   Ao    -A_,  =  ALi, 

A2-Ai  =  AV   A_i-A_2  =  A'_2, 

etc.,       etc.  ; 
that  is,  the  series 

A'      A'      A^        A' 

represents  the  differences  between  the  numbers  of  the  first 
series  of  differences  taken  in  order ;  it  is  called  the  second 
series  of  differences.     Likewise, 

A^'      A^/      A''        A'' 

Li     J,    Li    Q,    lA    _j,    Li     _2, 

represent  the  differences  between  the  successive  numbers 
of  the  second  series  of  differences,  and  we  have  the  third 
series  of  diff'erences.  In  like  manner,  we  can  proceed  to 
ioviwWiQ  fourth.,  fifths  and  higher  series  of  differences^  but  the 
series  beyond  the  third  are  used  very  rarely. 

We  now  proceed  to  illustrate  the   above  formula  by  an 

example.     It  is  required  to  find  the  value  of  -^  at  100°  C. 

du 

from  the  values  given  by  Wiebe  *  for  the  vapor  tension 
p  of  water  at  the  temperature  6.  We  find  in  his  tables  the 
following  values  oi  p  and  6  from  0°.5  to  0°.5 : 

*  Tafeln  iiber  die  Spannkraft  des  Wasserdampfes,  Braunschweig,  1894. 


392 

CALCULUS 

[Ch.  XII. 

e 

P 

A 

A' 

A" 

99.0 

(733.24)^ 

99.5 

(746.52)_j 

(13.28)_, 

(0.20)_, 

100.0 

(760.00), 

(13.48)_i 

(0.21)_. 

(+0.01)_, 

100.5 

(773.69)^1 

(13.69)„ 

(0.20)„ 

(-0.01)_i 

101.0 

(787.58),2 

(13.89)i 

We  have  attached  the  proper  indices  to  the  numbers  in 
the  above  table,  and  by  substitution  in  equation  (1)  we  have 

dp^J_  fl3.48  + 13.69  _  1  0.01  -  0.01 1      ^-r  .  ^  . 

de     0.b\  2  6  2  J      "  '      ' 

that  is,  in  the  vicinity  of  100°  C,  an  increase  or  decrease 
of  pressure  amounting  to  27.17  mm.  of  mercury  corresponds 
to  a  rise  or  fall  in  temperature  of  one  degree. 

We  now  pass  to  the  proof  of  our  formula,  employing  the 
same  quantities  as  in  the  example.     Let 

(3)  p  =/((9)  =  ^  +  ^(9  +  (7(92  +  i)(93  +  JEO^  4. ... 

be  a  series  representing  the  pressure  jt?  as  a  function  of  6. 
Its  differentiation  gives 

(4)  ^  =fiO)  =  ^  +  2  (7^  +  3  D^  +  4  J£'<93  +  ..., 
du 

and  it  is  now  required  to  represent  the  coefficients  of  the 

series  in  terms  of  the  numerical  data  of  the  table.     These 

data  give  the  values  of  p  for  the  temperatures 

(9,  e±h,  0±2h. 
If  in  equation  (3)' we  substitute   6 -\- h  and  0  —  h  for  6, 
we  obtain  the  following  equations : 
Pi=A0  +  h}    . 

=  A  +  B(e  +  A) + 0(0  +  hy  +  Did  +  hy  +  E(e  +  ny-, 

(5) 
p_,=fC0-h} 

=  A  +  B{0-  h)+c(0  -  hy  +  D(0  -  hy  +  Ei0-  hy-. 


1.]  DIFFERENTIATION  OF  FUNCTIONS  393 

If  we  expand  the  binomials  in  the  parentheses,  and  sub- 
tract equation  (3),  we  have 

p^  -p^  =Bh+  (7(2  dh  +  ¥)  +  i>(3  (92A  +  3  OW  +  ¥) 
+  J^(4  e^h  +  6  (92A2  4_  4  e¥  +  AO  +  ..., 

Pq~P-^  =  Bh  +  0(2 Oh  -  ^2)  +  i)(3 (92^-3 (9^2  +  ^3) 
+  EQi  e%  -  6  (92^2  +  4  (9^3  _  ^4)  _^  .... 

By  adding  these  equations  together  and  remembering  that 

Pi-Po  =  ^0  and  ^0  -  p_i  =  A_i, 
we  have 

Ao  +  A_i  =  2  m  +  2  (7  •  2  6^^  +  2  i> .  (3  (92A  -I-  A^) 

or,  after  dividing  by  2  A, 

1  .  A^  +  A_^  ^  ^  _^  2  (76>  -^SBO^h  +  4^6>3  4-  ... 
4-i>A2_^4^5l^2_^.... 

The  first  line  of  the  right  member  is  the  derivative  as 
developed  in  equation  (4),  so  that  we  may  put 

(6)  ^  =  l(^^ii±^)-(2)A2  +  4^^A2+...). 

If  h  be  made  very  small,  the  terms  into  which  P  enters 
as  a  factor  will  usually  be  so  small  that  if  they  are  omitted 
we  still  have  an  approximate  equality,  i.e,  we  have  approxi- 
mately 

(7)  j^^l/A  +  AA 
^  ^  d0     h\       2      J 

We  have  thus  derived  the  first  term  of  our  general  for- 
mula, equation  (2),  by  assuming  that  we  should  still  have 
the  desired  degree  of  approximation  if  we  neglect  the  series 

in  equation  (6). 


394  CALCULUS  [Ch.  Xll. 

If  this  should  not  be  the  case,  a  closer  approximation  can 
be  obtained  as  follows : 

In  equation  (3)  we  substitute  6  -\-2h  or  6  —  2h  for  ^,  and 
thus  get 

J92    =  tI  +  ^(i^  +  2  A)  +  6'(6>  +  2  hy  +  D(0  +  2  hy 

+  E(^e  +  2hy^-"', 

p_^  =  A-{-B(0-2h)-{-  0(6  -2hy^-D(e-2hy 

-^£Xe-2hy+"'. 

Subtracting  these  equations  from  equations  (5),  and  keep- 
ing in  mind  that 

P2-Pi  =  ^1  and  p_^  -  ;?_2  =  A_2, 
we  find,  after  some  simple  reductions,  that 

Ai  =Bh+  (7(2  eh  A-  3  h^)  +  i>(3  e^h  +  9  (9^2  +  7  A^) 
+  -£;(4  (93A  +  18  e^h?  +  28  e¥  +  15  A*)  +  -, 

A_2  =  Bh  +  (7(2  eh-n  ¥)  +  i>(3  (92^-9  dW  +  7  A^)- 
+  ^(4  e^h  -  18  (92/^2  +  28  e¥  -  15  A*)  +  .... 

We  have  previously  found  (p.  393) 

Aq    =m  +  (7(2(9A  +  A2)  +  i>(3  6>2A  +  3l9A2  +  A3) 
+  J^(4  (93A  +  6  ^2/^2  +  4  (97^3  +  ;^4^)  _|_  ... . 

A_i  =  Bh  +  0(2  Oh  -¥)  +  D  (3  O'^h  -  S  0h^  +  W) 
+^(4  e^h  -  6  (92^2  ^  4  51^3  _  7,4)  _!_.... 

and   subtract  these  four   equations   from    one  another ;    in 
accordance  with  the  notation  introduced  on  p.  391,  we  have 

A'=  (7-  2  A2  4-i)(6  (9^2+6  h^^+E(12  mt^+24:  Oh^+U  ¥) 

+  •••; 
A'_i=  O-  2  h^-hl)(6  (9A2)  +  ^(12  (92^2+2  A4)  +  -  ; 

A'_2=  (7.  2  A2+i>(6  (9A2_6  A3)+jE^(12  (92^2-24  (9A3+14  A*) 

+  ••• 


1-2.]  DIFFERENTIATION  OF  FUNCTIONS  395 

From  these  still  another  subtraction  gives  us  for  the  new 
differences  A%  =  A' -  ALj  and  A^'_2  =  AL^  -  AL2  the 
values 

A"_i  =  I)  '  Q¥  +  ^(24  e¥  +  12  ¥)  +  ... ; 

A"_2  =  i> .  6  A3  -f  ^(24  (9^3  _  12  A*)  +  .... 
On  adding  these  two  expressions,  we  obtain 

A'Li  +  A''_2  =  2  i>  .  6  7i3  +  2  JS' .  24  6>A3  -I-  ... ; 
or  finally,  after  dividing  by  2  A, 

i  ,  ^  -1  +  ^  -2  ^  6  2)^2  ^  24  jE'6>A2  +  ... 

=  6  (i>A2  +  4  ^6>A2 +...)+.... 

But  the  right  member  of  this  equation  is  just  the  one  whose 
value  we  have  been  seeking.  If  we  neglect  the  remaining 
right-hand  terms  and  substitute  the  approximate  *  value 
thus  found  in  equation  (6),  we  have 

(8)  jg^l|A,  +  A_,_lA%  +  A%) 

^  ^  dO     h\        2  6  2  ) 

as  a  better  approximation  to  the  required  derivative. 
A  still  more  exact  formula  is  the  one  given  above, 

^P  -  1  [ Aq  +  A_,      1  A^L,  +  A^^_2      1  A^%  +  A^%) 
de     hi        2  6  2  30  2  y 

where  A^^_2  and  A-^_3  have  meanings  entirely  analogous  to 
those  of  the  foregoing  quantities,  and  the  formula  is  proved 
in  a  similar  way. 

Art.  2.  Integration.  Oftentimes  it  is  necessary  to  calcu- 
late the  value  of  the  definite  integral 


ydx 


*  Approximate,  because  if  the  terms  F^  -i-Gd^  +  •••  were  taken  into  con- 
sideration, we  should  have  additional  terms  multiplied  by  A*,  h^,  •••. 


396  CALCULUS  [Ch.  XII. 

from  the  data  of  a  table.  To  this  end  we  may  either  inte- 
grate a  suitable  interpolation  formula  which  conforms  suffi- 
ciently closely  to  the  observed  values,  or  we  may  plot  the 
curve  and  determine  the  area  representing  the  value  of  the 
integral.  (See  p.  253.) 
If  for  the  values  of  x^ 


sufficiently  close  together,  there  are  known  corresponding 
values  of  ?/, 

l/o^    Vv    Vv     ^3'     "*'     Vnt 

the  required  integral  Q  may  as  a  first  approximation  be  put 

equal  to 

(1)  ^  =  (^j-^„)I^L  +  ll +  (,:,_  ^^)  ^1+1?  +  .. . 

this  formula  gives  the  sum  of  the  areas  of  the  trapezoids 
formed  by  the  axis  of  abscissas,  any  two  neighboring  y-co- 
ordinates  and  the  lines  connecting  their  extremities ;  evi- 
dently it  should  be  used  only  when  these  connecting  lines 
approximate  sufficiently  near  to  the  curve. 

Example.  At  the  time-intervals  t^^  t^,  to,  •••,  t^,  let  the  strength  of  an 
electrical  current  be  found  to  have  the  values  Cq,  Cj,  Cg,  •••,  C„.  If  the 
current  passed  through  a  silver  solution,  the  amount  of  silver  precipi- 
tated is  equal  to  the  product  of  the  equivalent  weight  of  the  metal  by 
the  quantity  of  electricity  E  (electro-chemically  measured)  which  has 
passed  through  the  circuit.     But  the  theory  of  electricity  shows  that 

E=\     Cdt; 


from  the  observed  values  we  obtain  accordingly  the  following  approxima- 
tion to  the  value  of  E : 

E  =(t,  -  to)  ^i^±^  +  (t,  -  t,)  £l±^  +   ...  +(tn  -  4,_i)   <^n-l+Cn, 


2.]  DIFFERENTIATION  OF  FUNCTIONS  397 

A  closer  approximation  is  secured  by  passing  a  parabola  through 
every  three  consecutive  extremities  of  the  ordinates.  To  find  the 
parabola,  for  instance,  which  passes  through  the  ends  of  the  ordinates 
whose  values  are  i/q,  y^,  y^,  we  consider  the  curve 

(2  a)  y  =f{x)  =  ^0  +  «  (^  -  ^o)  +  &  (^  -  ^o)^- 

This  curve  passes  through  the  point  (ar^,  y^),  as  is  seen  by  direct  sub- 
stitution of  these  values.  It  will  pass  through  the  points  (x^,  y^)  and 
('^2'  ^2)  ^^^Oy  if  a  and  b  are  so  determined  that 

(2)  y^  =  y^-\-a  (x^  -  a:o)  +  6  (x^  -  x^y, 

(3)  2/2  =  ^0  +  «  (-^2  -  ^0)  +  *  (^2  -  ^o)^- 

By  solving  these  equations  we  find  the  values  of  a  and  b  (expressed  in 
terms  of  known  quantities),  which  must  be  used  in  equation  (2  a),  in 
order  that  the  resulting  curve  may  pass  through  the  three  given  points. 

The  curve  is  recognized  as  a  parabola,  because  by  transformation  of 
coordinates  (see  pp.  59  et  seq.)  its  equation  can  be  brought  into  the  form 
rj=2pi^.     We  accomplish  this  by  putting 

x  =  $+a,    y  =  rj  +  l3, 

and  giving  such  values  to  a  and  p  that  in  the  new  equation  the  constant 
term  and  the  coefficient  of  i  vanish.  We  observe  also  that  the  $-  and 
ry-axes  are  parallel  to  the  x-  and  y-Sixes,  so  that  the  portion  of  a  parabola 
passing  through  the  points  1,  2,  3  belongs  to  a  parabola  whose  axis  is 
parallel  to  that  of  X.  It  can  be  proved  that  a  parabola  is  fully  deter- 
mined by  three  of  its  points  and  the  direction  of  its  axis. 
The  solution  of  equations  (2)  and  (3)  gives 

^4)  a  =  (yi-  yo)(^2  -  ^0)^  -  (1/2  -  yo)(^i  -  ^o)'^^ 

(^1-2:0)  (^2-^0)  (^2-^1) 
^gx  J  ^  (^2  -  yp)  (^1  -  ^0)  -  (.Vi  -  -Vo)  (^2  -  ^0)^ 

and  integrating  y  with  respect  to  x  between  the  limits  Xf^  and  x^,  we  have 
(6)  f  V  dx  =  2/0  (x^  -  Xq)  +  ^  (^2  -  ^0)^  +  o  (^2  -  ^0)^ ; 

where  a  and  b  have  the  values  given  in  equations  (4)  and  (5). 


398  CALCULUS  [Ch.  XII. 

We  may  treat  in  a  similar  manner  x^,  x^,  x^,  and  the  corresponding 
values  1/2,  ?/3,  y^y  respectively ;  then  x^,  x^,  Xq,  and  ij^,  ?/-,  ^g,  and  so  on.  If 
n  be  even,  we  obtain  the  values  of  a  series  of  integrals  terminating  with 

ydx, 

and  the  value  of  the  required  integral  is 

^  =  I    y  dx  +  \    y  dx  +  '-'-[-  \       ydx. 

Jx^  J  x^  Jxn-2 

If  n  be  odd,  an  additional  pair  of  values  of  x  and  y  may  be  determined 
by  observation  or  by  interpolation,  or  we  can  also  in  one  case  compute 
the  area  of  the  surface  comprehended  between  two  successive  coordinates 
as  a  trapezoid. 

If  the  distances  between  the  ordinates  are  all  the  same,  so  that 

X^         •''O  ~~  "^^2         a^j  :=  •••  Xn         ^n— 1  -—  '^y 

the  above  equation  may  be  simplified,  and  assumes  the  form 

^  =  3  [^0  +  ^n  +  y{yi  +  2^2  +   -  +  Vn-l)  +  2  (^2  +  2/4  +   -  +  yn-2)], 

an  expression  known  as  Simpson's  Formula  (of  course  n  is  still  an  even 
number).  By  planning  the  observations  (p.  396)  so  that  this  formula  can 
be  used,  much  labor  of  computation  may  be  avoided.  We  accomplish 
this  in  the  above  example,  for  instance,  by  reading  the  current  strength 
at  equal  time  intervals. 

Numerical  Example.     Suppose  we  have  given  the  corresponding  values 
Xq  =  1.000,  y^  =  0.5000, 

x^  =  1.500,  y^  =  0.3077, 

3:2  =  2.000,  3/2  =  0.2000, 

and  it  is  required  to  find  the  value  of 

e=  \    ydx. 

Formula  (1)  gives  us,  as  an  approximate  value  of  /, 

g  ^  0  5  0.5000  +  0.3077      ^^  0.3077  +  0.2000  ^ 
2  2 


2.]  DIFFERENTIATION   OF  FUNCTIONS  399 

If  we  use  formula  (6)  instead,  we  may  expect  a  closer  approximation  ; 
we  find,  in  fact, 

2^0  (^2  -^i)  =  +  0.5000 

|(^2-^i)  =-0.2346 
|(^2-^o)'=  + 0.0564 


e        =     0.3218 
Since,  in  the  example, 

we  can  apply  the  simpler  formula  (7)  more  conveniently,  and  thus  obtain 

0  =  ^  [0.5000  +  4  X  0.3077  +  0.2000]  =  0.3218, 
o 

a  value  which,  from  the  nature  of  the  case,  must  be  exactly  the  same  as 
that  obtained  from  equation  (6). 

In  order  to  test  the  closeness  of  our  approximation,  the  values  of  y  in 
this  example  were  not  determined  by  observation,  but  from  the  equation 

so  that  knowing  exactly  the  relation  between  x  and  y,  we  could  deter- 
mine accurately  for  comparison  the  value  of  the  required  integral.  It  is 
obtained  by  integrating,  with  the  result 

0  =  r_^?^  =  arc  tan  2  -  arc  tan  1  =  0.321751. 

^1  1  +  X2 

We  see,  therefore,  that  equations  (6)  and  (7)  give  quite  close  approxi- 
mations. 


27 


APPENDIX 

COLLECTION  OF  FORMULAE 

1.   a'^'or  =  «"+'•.  2.    {a^'Y  =  a"'-. 

3.  (1  +  X)-  =  1  +  nx  +  !^^!Llll)  :,2  ^  K^  -  1^)0^  ^  -)  :.3  _^  ... 

(Binomial  Series.)         +  K^  -  1)  -  (^  -  ^  +  1)^.  ^  .... 

The  three  formulae  above  hold  for  all  values  of  n,  positive  or  negative, 
integral  or  fractional,  and  for  x  (in  the  third)  numerically  less  than  unity. 

4.  n!  =  1.2.3.4...  (n  -  l)n.  «    i       n/-      ii 

^  ^  8.   log  -^a  =  -  log  a. 

5.  log  a6  =  log  a  +  log  b.  ^ 

^  9.   log  1  =  0. 

6.  log  -  =  log  a  —  log  b. 

b  1 

«    X  1  10.   log  -  =  -  log  a. 

7.  tog  a"*  =  n  log  a.  a 

Formulae  (5)  to  (10)  hold,  whatever  the  base. 

11.  sin  (a:  +  2  riTr)  =  sin  a:.  to  /tt  ,      \ 

^  ^  18.   cos(  -  +  ^1  =  —  sin  a:. 

12.  cos  (x  +  2  rnr)  =  cos  a;.  \-         / 

13.  sin  (1  -  ^)  =  cos  X.    .  ^^'  t^"  (2  +  '')  ^  ~  ^^*  '^' 

14.  cos  ^1  -  x)  =  sin  X.  20.  cot  ^^  +  x  j  =  -  tan  x. 

15.  tan  (1  -  x)  =  cot  x.  ^1.  sin  (tt  -  x)  =  sin  a:. 


16.   cot  l-  —  x]  =  tan  a;. 

\2     ; 


22.  cos  (tt  —  x)  =  -  cos  X. 

23.  tan(7r  —  a:)  =  —  tan  x. 
17.   sinf- +  a:J  =  cosa;.                         24.  cot (tt  -  a;)  =  -  cot  a:. 

401 


402 


APPENDIX 


25.  sin  (  —  x)  =  —  sin  x ;  cos  (  —  x)  =  cos  x. 

26.  tan  (  —  x)  =:  —  tan  x ;  cot  (  —  x)  —  —  cot  x. 

__     .  sin  a:        ,  cos  x 

27.  tan  a:  = ;  cotx  =  ^ 

cos  X  sin  X 

28.  sin^x  +  cos^j;  =  1. 

29.  sin  X  =  Vl  —  cos'^x ;  cos  x  =  Vl  —  sin^ar. 
tanx 


30.  sin  X  = 

31.  cos  X  = 


Vl  4-  tan 2a;' 

1 


Vl  +  tan^x 

32.  sin  (x  +  ?/)  =  sin  a:  cos  y  +  cos  a:  sin  y. 

33.  sin  (^x  —  y)=  sin  a:  cos  ?/  —  cos  x  sin  ^. 

34.  cos  (x  4-  //)  =  cos  X  cos  ?/  —  sin  x  sin  y. 

35.  C03  {x  —  y)=  cos  a:  cos  ?/  +  sin  x  sin  y. 

36.  sin  2  X  =  2  sin  a:  .cos  x. 

Zl.  cos  2  a:  =  cos^a:  —  sin^a:  =  2  cos^x  —  1  =  1—2  sin%c 

38.  sin  X  =  2  sin  -  cos  -• 

2        2 

39.  cos  X  =  2  cos2-  -1  =  1-2  sin2 ?. 

2  2 


40.   sin  X  +  sin 


2sin^±lcos^^:ii^. 


41.   sm  X  —  sm  w  =  2  cos  — '-^  sin ^• 

^  "2 


t4.   tan  (x  +  ?/)  = 

45.  tan  (x  —  y)  = 

46.  tan  2  x  = 


42.  cos  X  +  cos  y  =  2 

43.  cos  X 
tan  X  +  tan  y 


2 

,x  + ?/ 


cos 


cos  y  =  —  2  sin  — ^^Jl  sin 


X  -  y. 
2 

X  —  y 


1  —  tan  X  tan  y 

tan  X  —  tan  y 
1  +  tan  X  tan  ?/ 

2  tan  X 


48.   sin 


Jl 

+  cosx 

-\ 

2 

Jl 

—  cos  X 

-\ 

2 

1  -  tan^a; 


yift      4.        ^        ^1   —  cosx 

49.   tan-=\- • 

2        ^  1  +  QOS  X 


APPENDIX  403 


For  arithmetical  series, 


50.    Sn  =  l(2a+ln-  l]d).  51.    1  +  2  +  ...  +  n  =  ^(^  +  ^X 

For  geometric  series, 

52.  S„^«('"-l)  =  «(l-'"). 

r  —  1  1  —  r 

53.  12  +  02  +  32  +  ...  +  n2  =  ^(^  +  l)(2n  +  l) 

6 

54.  Area  of  triangle  (base  =  b;  altitude  =  //)  :  |  bH. 

55.  Area  of  triangle  (y  =  angle  included  by  sides  a  and  b):  1  ao  sm  y. 

56.  Area  of  parallelogram  (base  =  b;  altitude  =  H):  bH. 

57.  Area  of  parallelogram  (y  =  angle  included  by  sides  a  and  b) :  ab  sin  y. 

58.  Area  of  trapezoid  (bases  =  b^  and  ftgi  altitude  —  H):  -^r — ?  ^. 

59.  Circumference  of  circle  (radius  =  r) :  2  irr. 

60.  Area  of  circle  :  irr^. 

61.  Area  of  sector  of  circle  (number  of  radians  in  arc  =  ^)  :  |  <l>r^. 

62.  Area  of  ellipse  (semi-axes  =  a  and  b)  :  «&7r. 

63.  Volume  of  prism  (area  of  base  =  A  ;  altitude  =  H):  AH. 

64.  Volume  of  pyramid  (area  of  base  ==  A  ;  altitude  =  H):\  AH. 

65.  Volume  of  cylinder  (radius  =  r ;  altitude  =  H)  :  irr^H. 

66.  Surface  of  cylinder  :  2  tttH. 

67.  Volume  of  cone  :  \  icr^H. 

68.  Convex  surface  of  cone  (radius  =  r ;  side  —  s)  :  irrs. 

69.  Convex  surface  of  frustrum  of  cone  (radii  =  r  and  p;  side  ==  s) ; 
(r  +  p)  7r6-. 

70.  Volume  of  sphere  :  |  irrK 

71.  Surface  of  sphere  :  4  Trr^. 

In  the  quadratic  equation 

ax"^  -}-  bx  -{■  c  =  Of 
the  following  relationships  exist : 

If  &2  _  4  ac  is  the  roots  are 

72.  i.     positive,  real  and  unequal ; 

73.  ii.    zero,  real  and  equal ; 

74.  iii.  negative,  imaginary. 

^  OF  THE 
iiMiv/CDQlTY 


INDEX 


Abscissa,  definition  of,  8. 
Abscissae,  axis  of,  9. 
Acceleration,  definition  of,  279. 
Algebraic  points,  real  and  imaginary, 

75. 
Analytic      Geometry,      fundamental 

principle  of,  1.2. 
Angle,  coordinate,  9. 
Archimedes,  spiral  of,  74, 
Artifices,  special,  in  integration,  195. 
Asymptote,  derivation  of  word,  62. 
Asymptotes    of    hyperbola,    59,    65, 

306. 
Attraction  of  rod  on  point,  218. 
Auxiliary  circle  to  ellips;e,  304. 
Axis  of  abscissae,  9. 

of  coordinates,  9. 

major  and  minor,  of  ellipse,  45. 

of  ordinates,  9. 

real  and  imaginary  of  hyperbola, 
57. 

of  symmetry  of  parabola,  22. 

Barometric  readings  reduced  to  0°  C. , 

358. 
Base  of  logarithms,  136.  ^ 
Bernoulli,  147,  299. 
Binomial  theorem,  341. 
Boyle's  law,  2. 
Briggean  logarithms,  139. 

Calculation  with  small  quantities,  357. 

Cane  sugar,  solubility  of,  5. 

Carbon  monoxide,  heat  of  combustion 

of,  239. 
Cartesian  geometry,  7. 


Catenary,  268. 

Chemical  reactions,  partial,  242. 

total,  240. 
Circle,  auxiliary  to  ellipse,  51. 

equation  of,  16. 
Circular  functions,  148. 

measure,  74. 
Coefficient  of  expansion,  105. 
Comet,  determination  of  orbit,  73. 
Common  logarithms,  139. 
Concavity  of  curves,  276. 
Concentration,  definition  of,  240. 
Conic  sections,  71. 
Constant,  77. 

derivative  of,  128. 
Constant  of  integration,  174. 

geometric  signification  of,  176. 

physical  signification  of,  180. 
Construction  of  a  point,  10. 
Continuity,  definition  of,  164. 

of  curves,  160. 
Continuous  quantity,  97. 
Convergence      of      series,      general 
theorems  on,  314. 

rapidity  of,  322. 
Convergent  series,  312. 
Convexity  of  curves,  276. 
Cooling,  Newton's  law  of,  224. 
Coordinate  paper,  4. 
Coordinates,  axes  of,  9. 

polar,  69. 

rectangular,  9. 

transformation  of,  63. 
Criteria  for  forms  of  curves,  table  of, 

368. 
Cross-section  paper,  4. 


405 


406 


INDEX 


Current    strength,   measurement  of, 

388,  396. 
Curve,  equation  of,  12. 
Curves,     criteria    concerning     their 
forms,  368. 
quadrature  of,  249. 
rectification  of,  267. 
Cycloid,  269. 

Definite  integrals,  245. 

laws  of  operation  of,  262. 

limits  of,  253. 
Degree  of  homogeneous  functions,  298. 
Dependent  variable,  78. 
Derivative,  definition,  117. 

discontinuous,  164. 

of  constant,  128. 

geometric    interpretation    of    the 
sign  of,  124. 

physical  signification  of,  109. 

of  power  with  any  exponent,  155. 

of  product,  130. 

of  quotient,  132. 

of  sin  X  and  cosaj,  121. 

of  sums  and  differences,  126. 

of  sc«,  120. 
Derivatives  of,  second,  272. 

general  rule  for  the  formation  of, 
115. 

higher,  273. 

partial,  284. 
Descartes,  7. 

Differences,  derivative  of,  126. 
Differentiation,  definition  of,  118. 

of  empiric  functions,  389. 

of  logarithmic,  157. 
Directrix  of  the  ellipse,  50. 

of  the  hyperbola,  57. 

of  the  parabola,  21. 
Discontinuity,  160. 
Discontinuous  functions,  162. 
Dissociation,  definition  of,  233. 
Divergent  series,  312. 

Eccentricity  of  ellipse,  44. 

of  hyperbola,  59. 
Electricity,  transmission  of,  303. 


Elements  of  growth  of  United  States,  1. 
Ellipse,  auxiliary  circle  of,  51. 

axes  of,  45. 

definition  of,  42. 

directrix  of,  50. 

eccentricity  of,  53. 

equation  of,  42. 

equation  of,  in  polar  coordinates, 
72. 

focal  properties  of,  303. 

foci  of,  45. 

form  of,  44. 

problems  on  directrix  of,  53. 

quadrature  of,  255. 

semi-axes  of,  45 

vertices  of,  45. 
Empiric  functions,  differentiation  of, 
389. 

integration  of,  395. 
Epsilons,  definition  of,  91. 

properties  of,  92. 

use  of,  229. 
Equation,  of  a  circle,  16. 

of  a  comet's  path,  73. 

of  a  curve,  12. 

of  the  ellipse,  42. 

of  the  ellipse  in  polar  coordinates, 
71. 

general,  of  the  first  degree,  27. 

general,  of  the  nature  of,  35. 

of  a  hyperbola,  56. 

of  a  hyperbola  in  polar  coordinates, 
71. 

linear,  27. 

of  normal,  34. 

of  a  parabola,  20. 

of  a  parabola  in  polar  coordinates, 
71. 

of    a    straight    line    through    the 
origin,  22. 

of  any  straight  line,  24. 

symmetric,  31. 

Van  der  Waals',  m. 
Equilateral  hyperbola,  59. 
Equivalent  weight  of  barium,  386 

weight  of  sodium,  385. 
Error,  percentage  of,  385. 


INDEX 


407 


Errors,  absolute  and  relative,  385. 

estimation  of,  383. 
Estimation  of  errors,  383. 
Euler,  299. 

Euler's  theorem  of  homogeneous  func- 
tions, 299. 
Expansion,  coefficient  of,  105. 

of  rod,  105. 

speed  of,  107. 

Flame,  temperature  of,  236. 
Focal  properties  of  ellipse,  303. 

of  parabola,  301. 
Focal  ray  of  ellipse,  305. 

of  parabola,  302. 
Foci  of  ellipse,  45,  306. 

of  hyperbola,  57. 
Focus  of  parabola,  21,  303. 
Formula,  hypsometric,  220. 
Fonuulae,  fundamental,  of  the  Integral 
Calculus,  176. 

of  reduction,  199. 

collection  of,  400. 
Function  concept,  110. 
Function,  definition  of,  110. 

explicit,  296. 

homogeneous,  298. 

homogeneous,  degree  of,  298. 

implicit,  296. 
Functions,  circular,  148. 

differentiation  of  implicit,  295. 

discontinuous,  162. 

empiric,  differentiation  of,  389. 

empiric,  integration  of,  395. 

Euler's  theorem  of  homogeneous, 
299. 

examples  of,  from  nature,  113. 

exponential,  143. 

of  functions,  150. 

inverse  trigonometrical,  147. 

of  several  variables,  272. 

Gauss,  346. 

Gay  Lussac's  Law,  29. 

General  equation,  the  nature  of,  35. 

Geometric  Series,  313. 

Graph,  15. 


Graphic  representation,  1. 
Gravitation,  Newton's  Law  of,  218. 
Gregory,  344. 
Gregory's  Series,  344. 
Growth,    elements    of,     for    United 
States,  1. 

Harmonic  Series,  313. 
Heat  of  combustion  of  carbon  mon- 
oxide, 239. 
Hertz,  303. 
Homogeneous  functions,  298. 

degree  of,  298. 

Euler's  theorem  of,  299. 
Horstmann,  389. 
Huygens,  281. 
Hyperbola,  asymptotes  of,  306. 

definition  of,  55, 

eccentricity  of,  59. 

equation  of,  56. 

equation  of,  in  polar  coordinate, 
72. 

equilateral,  69. 

form  of,  57. 

foci,  57. 

imaginary  axis  of,  57. 

quadrature  of,  255,  266. 

real  axis  of,  57. 

vertices,  57. 
Hyperbolic  logarithms,  258. 
Hyperboloid  of  revolution,  267. 
Hypo-cycloid,  269. 
Hypsometric  formula,  220. 

simplified,  359. 

Identity  symbol,  295. 
Imaginary  algebraic  point,  75. 
Imaginary  curves,  76. 
Implicit  functions,  296. 
Indefinite  integral,  253,  269. 
Independent  variable,  77. 
Indeterminate  forms,  347. 

limits  of,  350. 

types  of,  355. 
Infinity,  85. 

Inflexion,  point  of,  277,  364,  366,  369. 
Inflexional  tangent,  365. 


408 


INDEX 


Integrals,  definite,  245. 

indefinite,  253,  269. 

notation  of,  171. 

preliminary  table  of,  176. 

table  of,  214. 
Integration,  constant  of,  174. 

definition  of,  170. 

by  decomposition  into  partial  frac- 
tions, 203. 

of  empiric  functions,  395. 

geometric  signification  of  constant 
of,  176. 

by  inspection,  200. 

by  introduction  of  new  variables, 
186. 

by  parts,  191. 

physical  signification  of  constant 
of,  180. 

by  series,  342. 

special  artifices  in,  195. 

of  sums  and  differences,  184. 

by  transformation  of  function,  196. 
Intensity  of  heat,  minimum  of,  374. 
Intercept,  29. 

Interest,  computation  of,  145. 
Inverse  trigonometrical  function,  147. 
Inversion  of  sugar,  183,  270. 

Lactone,  definition  of,  243. 
Law,  Boyle's,  2. 

of  cooling,  224. 

Gay  Lussac's,  29. 

of  gravitation,  218. 

of  Mass  Action,  167. 
Leibnitz,.  97. 
Leibnitz's  theorem,  333. 
Limit,  definition  of,  80. 

rigorous  definition  of,  81. 
Limits  of  definite  integrals,  253. 

fundamental  theorems  of,  87. 

illustrations  of,  79. 

propositions  concerning,  90. 
Linear  equation,  27. 
Locus,  15. 
Logarithmic  differentiation,  157. 

functions,  136. 

series,  337. 


Logarithms,  Briggean,  139. 

common,  139. 

definition,  136. 

hyperbolic,  258.      • 

Napierian,  139. 

natural,  139. 

notation  of,  140. 

relations  between   with    different 
bases,  146. 
Long's  data  for  inversion  of  sugar, 
183. 

Maclaurin,  325. 
Maclaurin's  theorem,  325. 
Marconi,  303. 
Mass  Action,  law  of,  167. 
Mass  of  a  rod  of  varying  density,  261. 
Maxima  and  minima,  conditions  for, 
364. 

definition  of,  362. 

examples  of,  368. 
Minimum  of  intensity  of  heat,  374. 
Mortgage,  its  analogy  with  the  gen- 
eral equation,  35. 
Motion  of  freely  falling  body,  102. 
Motion,  oscillatory,  280. 

on  a  parabola,  99. 

pendular,  281. 

Napier,  139. 

Napierian  logarithms,  139. 

Natural  logarithms,  139. 

Newton,  97. 

Newton's  law  of  cooling,  224. 

of  gravitation,  218. 
Normal  to  ellipse,  305. 

to  parabola,  302. 
Normal  equation  of  the  straight  line, 

34. 
Notation  of  sums,  248. 

Oblate  spheroid,  266. 
Observation,  weight  of,  387. 
Ordinate,  definition  of,  8. 
Ordinates,  axis  of,  9. 
Origin  of  coordinates,  9. 
Oscillatory  motion,  280. 


INDEX 


409 


Parabola,  definition  of,  14. 

equation  of,  20. 

equation  of,  in  polar  coordinates, 
71. 

focal  properties  of,  301. 

focal  ray  of,  302. 

motion  on,  99. 

normal  to,  302. 

parameter  of,  20. 

quadrature  of,  245. 

semi-cubical,  268. 

vertex  cf,  302. 
Paraboloid  of  revolution,  volume  of, 

260. 
Parameter  of  parabola,  20. 
Partial  derivatives,  284. 

derivation,  higher,  288. 

derivation,  notation  of,  286. 
Pendular  motion,  281. 
Percentage  of  error,  385. 
Plotting  a  curve,  10. 

a  point,  10. 
Point,  algebraic,  75. 
Point  of  inflexion,  277. 
Polar  coordinates,  69. 
Product,  derivative  of,  130. 

Quadrants,  9. 
Quadrature  of  curves,  249. 

of  ellipse,  255. 

of  hyperbola,  257,  266. 

of  parabola,  245. 
Quantity,  continuous,  97. 
Quotient,  derivative  of,  132. 

Radian,  74. 

Reactions,  partial  chemical,  242. 

total  chemical,  240. 
Real  algebraic  point,  75. 
Rectification  of  curves,  267. 
Reflection,  law  of,  376. 
Refraction,  index  of,  380. 

law  of,  378. 
Resistance,  measurement  of,  387. 
Revolution,  hyperboloid  of,  267. 

paraboloid  of,  260. 
Rotation,  direction  of,  22. 


Saturated  solution,  5. 
Second  derivative,  geometric  meaning 
of,  276. 

physical  interpretation  of,  278. 
Semi-axes  of  ellipse,  45. 
Semi-cubical  parabola,  268. 
Series,  binomial,  341. 

convergent,  312. 

for  cos  ic,  329. 

divergent,  312. 

for  e,  323. 

for  e^,  328. 

geometric,  313. 

Gregory's,  344. 

harmonic,  313. 

infinite,  310. 

integration  by,  342. 

logarithmic,  337. 

Maclaurin's,  325. 

practical  applicability  of,  314. 

for  sin  x,  329. 

sum  of  infinite,  311. 

table  of,  346. 

for  tan  x,  332. 

Taylor's,  334. 
Simpson's  formula,  398. 
Sine-curve,  124. 
Small  quantities,  357. 
Soap-bubble,  relation  between  pres- 
sure and  diameter,  4. 
Sodium,  equivalent  weight  of,  385. 
Solubility  curve,  6. 
Solubility  of  cane  sugar,  6. 
Solubility,  definition  of,  5. 
Specific  heat,  definition,  227. 
Speed,  102. 

Speed  of  reaction,  110,  168. 
Sphere,  volume  of,  259. 
Spheroid,  oblate,  266. 
Spiral  of  Archimedes,  74. 
Straight  line,  equation  of,  22. 

problems  on,  30." 
Sugar,  inversion  of,  183,  270. 
Sum  of  infinite  series,  311. 
Sums,  derivatives  of,  126. 

notation  of,  248. 
Symmetric  equation  of  straight  line,  31. 


410 


INDEX 


Table  of  criteria  for  forms  of  curves, 
369'. 

of  derivatives,  158. 

of  integrals,  214. 

of  series,  346. 

of  trigonometric  and  other  formu- 
Ise,  401. 
Tangent  to  curve,  45. 
'i'angent,  inflexional,  365. 
Taylor,  334. 

Taylor's  theorem,  334. 
Temperature  coefficient,  110. 

of  flame,  236. 

"^Jnited  States,  elements  of  growth  of,  1. 

Van  der  Waals'  equation,  QQ. 
Vapor  tension,  deflnition  of,  163. 

of  water,  392. 
Variable,  dependent,  78. 

independent,  77. 
Velocity,  102. 
Vertex  of  parabola,  302. 


Vertices  of  ellipse,  45. 

of  hyperbola,  57. 
Vibration  of  strings,  281. 
Volume  of  paraboloid  of  revolution, 
259. 

of  solid,  258. 

of  sphere,  259. 

Wave-motion,  281. 

Weight  of  observation,  387. 

Wheatstone  bridge,  387. 

Wiebe's   data   for  vapor  tension  of 
water,  391. 

Winkelmann's  data  for  law  of  cooling, 
227,  228. 

Wireless  telegraphy,  303. 

Work  done    in    expansion  of    com- 
pressed gas,  232. 
of  dissociating  gas,  233. 
of  perfect  gas,  230. 

X-axis,  10. 

Y-axis,  10. 


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